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<divclass="section" id="Accessing-Data-Along-Multiple-Dimensions-in-an-Array">
<h1>Accessing Data Along Multiple Dimensions in an Array<aclass="headerlink" href="#Accessing-Data-Along-Multiple-Dimensions-in-an-Array" title="Permalink to this headline"></a></h1>
<p>In this section, we will:</p>
<ulclass="simple">
<li><p>Define the “dimensionality” of an array.</p></li>
<li><p>Discuss the usefulness of ND-arrays.</p></li>
<li><p>Introduce the indexing and slicing scheme for accessing a multi-dimensional array’s contents</p></li>
</ul>
<p>We will encounter arrays of varying dimensionalities:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># A 0-D array</span>
<spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">(</span><spanclass="mi">8</span><spanclass="p">)</span>
<spanclass="c1"># A 1-D array, shape-(3,)</span>
<spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">([</span><spanclass="mf">2.3</span><spanclass="p">,</span><spanclass="mf">0.1</span><spanclass="p">,</span><spanclass="o">-</span><spanclass="mf">9.1</span><spanclass="p">])</span>
<spanclass="c1"># A 2-D array, shape-(3, 2)</span>
<spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">([[</span><spanclass="mi">93</span><spanclass="p">,</span><spanclass="mi">95</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">84</span><spanclass="p">,</span><spanclass="mi">100</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">99</span><spanclass="p">,</span><spanclass="mi">87</span><spanclass="p">]])</span>
<spanclass="c1"># A 3-D array, shape-(2, 2, 2)</span>
<spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">([[[</span><spanclass="mi">0</span><spanclass="p">,</span><spanclass="mi">1</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">3</span><spanclass="p">]],</span>
<spanclass="p">[[</span><spanclass="mi">4</span><spanclass="p">,</span><spanclass="mi">5</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">6</span><spanclass="p">,</span><spanclass="mi">7</span><spanclass="p">]]])</span>
</pre></div>
</div>
<p>Similar to Python’s sequences, we use 0-based indices and slicing to access the content of an array. However, we must specify an index/slice for <em>each</em> dimension of an array:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="gp">>>> </span><spanclass="kn">import</span><spanclass="nn">numpy</span><spanclass="k">as</span><spanclass="nn">np</span>
<spanclass="go"># A 3-D array</span>
<spanclass="gp">>>> </span><spanclass="n">x</span><spanclass="o">=</span><spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">([[[</span><spanclass="mi">0</span><spanclass="p">,</span><spanclass="mi">1</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">3</span><spanclass="p">]],</span>
<spanclass="gp">...</span>
<spanclass="gp">... </span><spanclass="p">[[</span><spanclass="mi">4</span><spanclass="p">,</span><spanclass="mi">5</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">6</span><spanclass="p">,</span><spanclass="mi">7</span><spanclass="p">]]])</span>
<spanclass="go"># get: sheet-0, both rows, flip order of columns</span>
<spanclass="gp">>>> </span><spanclass="n">x</span><spanclass="p">[</span><spanclass="mi">0</span><spanclass="p">,</span><spanclass="p">:,</span><spanclass="p">::</span><spanclass="o">-</span><spanclass="mi">1</span><spanclass="p">]</span>
<spanclass="go">array([[1, 0],</span>
<spanclass="go"> [3, 2]])</span>
</pre></div>
</div>
<divclass="section" id="One-dimensional-Arrays">
<h2>One-dimensional Arrays<aclass="headerlink" href="#One-dimensional-Arrays" title="Permalink to this headline"></a></h2>
<p>Let’s begin our discussion by constructing a simple ND-array containing three floating-point numbers.</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="gp">>>> </span><spanclass="n">simple_array</span><spanclass="o">=</span><spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">([</span><spanclass="mf">2.3</span><spanclass="p">,</span><spanclass="mf">0.1</span><spanclass="p">,</span><spanclass="o">-</span><spanclass="mf">9.1</span><spanclass="p">])</span>
</pre></div>
</div>
<p>This array supports the same indexing scheme as Python’s sequences (lists, tuples, and strings):</p>
<divclass="highlight-none notranslate"><divclass="highlight"><pre><span></span>+------+------+------+
| 2.3 | 0.1 | -9.1 |
+------+------+------+
0 1 2
-3 -2 -1
</pre></div>
</div>
<p>The first row of numbers gives the position of the indices 0…3 in the array; the second row gives the corresponding negative indices. The slice from <spanclass="math notranslate nohighlight">\(i\)</span> to <spanclass="math notranslate nohighlight">\(j\)</span> returns an array containing of all numbers between the edges labeled <spanclass="math notranslate nohighlight">\(i\)</span> and <spanclass="math notranslate nohighlight">\(j\)</span>, respectively:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="gp">>>> </span><spanclass="n">simple_array</span><spanclass="p">[</span><spanclass="mi">0</span><spanclass="p">]</span>
<spanclass="go">2.3</span>
<spanclass="gp">>>> </span><spanclass="n">simple_array</span><spanclass="p">[</span><spanclass="o">-</span><spanclass="mi">2</span><spanclass="p">]</span>
<spanclass="go">0.1</span>
<spanclass="gp">>>> </span><spanclass="n">simple_array</span><spanclass="p">[</span><spanclass="mi">1</span><spanclass="p">:</span><spanclass="mi">3</span><spanclass="p">]</span>
<spanclass="go">array([ 0.1, -9.1])</span>
<spanclass="gp">>>> </span><spanclass="n">simple_array</span><spanclass="p">[</span><spanclass="mi">3</span><spanclass="p">]</span>
<spanclass="go">IndexError: index 3 is out of bounds for axis 0 with size 3</span>
</pre></div>
</div>
<p>Given this indexing scheme, only <em>one</em> integer is needed to specify a unique entry in the array. Similarly only <em>one</em> slice is needed to uniquely specify a subsequence of entries in the array. For this reason, we say that this is a <em>1-dimensional array</em>. In general, the <em>dimensionality</em> of an array specifies the number of indices that are required to uniquely specify one of its entries.</p>
<divclass="admonition note">
<pclass="admonition-title fa fa-exclamation-circle"><strong>Definition</strong>:</p>
<p>The <strong>dimensionality</strong> of an array specifies the number of indices that are required to uniquely specify one of its entries.</p>
</div>
<p>This definition of dimensionality is common far beyond NumPy; one must use three numbers to uniquely specify a point in physical space, which is why it is said that space consists of three dimensions.</p>
</div>
<divclass="section" id="Two-dimensional-Arrays">
<h2>Two-dimensional Arrays<aclass="headerlink" href="#Two-dimensional-Arrays" title="Permalink to this headline"></a></h2>
<p>Before proceeding further down the path of high-dimensional arrays, let’s briefly consider a very simple dataset where the desire to access the data along multiple dimensions is manifestly desirable. Consider the following table from a gradebook:</p>
<tableclass="docutils align-default">
<colgroup>
<colstyle="width: 24%" />
<colstyle="width: 36%" />
<colstyle="width: 39%" />
</colgroup>
<thead>
<trclass="row-odd"><thclass="head"></th>
<thclass="head"><p>Exam 1 (%)</p></th>
<thclass="head"><p>Exam 2 (%)</p></th>
</tr>
</thead>
<tbody>
<trclass="row-even"><td><p>Ashley</p></td>
<td><p><spanclass="math notranslate nohighlight">\(93\)</span></p></td>
<td><p><spanclass="math notranslate nohighlight">\(95\)</span></p></td>
</tr>
<trclass="row-odd"><td><p>Brad</p></td>
<td><p><spanclass="math notranslate nohighlight">\(84\)</span></p></td>
<td><p><spanclass="math notranslate nohighlight">\(100\)</span></p></td>
</tr>
<trclass="row-even"><td><p>Cassie</p></td>
<td><p><spanclass="math notranslate nohighlight">\(99\)</span></p></td>
<td><p><spanclass="math notranslate nohighlight">\(87\)</span></p></td>
</tr>
</tbody>
</table>
<p>This dataset contains 6 grade-values. It is almost immediately clear that storing these in a 1-dimensional array is not ideal:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># using a 1-dimensional array to store the grades</span>
<spanclass="o">>>></span><spanclass="n">grades</span><spanclass="o">=</span><spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">([</span><spanclass="mi">93</span><spanclass="p">,</span><spanclass="mi">95</span><spanclass="p">,</span><spanclass="mi">84</span><spanclass="p">,</span><spanclass="mi">100</span><spanclass="p">,</span><spanclass="mi">99</span><spanclass="p">,</span><spanclass="mi">87</span><spanclass="p">])</span>
</pre></div>
</div>
<p>While no data has been lost, accessing this data using a single index is less than convenient; we want to be able to specify both the student and the exam when accessing a grade - it is natural to ascribe <em>two dimensions</em> to this data. Let’s construct a 2D array containing these grades:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># using a 2-dimensional array to store the grades</span>
<spanclass="o">>>></span><spanclass="n">grades</span><spanclass="o">=</span><spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">([[</span><spanclass="mi">93</span><spanclass="p">,</span><spanclass="mi">95</span><spanclass="p">],</span>
<spanclass="o">...</span><spanclass="p">[</span><spanclass="mi">84</span><spanclass="p">,</span><spanclass="mi">100</span><spanclass="p">],</span>
<spanclass="o">...</span><spanclass="p">[</span><spanclass="mi">99</span><spanclass="p">,</span><spanclass="mi">87</span><spanclass="p">]])</span>
</pre></div>
</div>
<p>NumPy is able to see the repeated structure among the list-of-lists-of-numbers passed to <codeclass="docutils literal notranslate"><spanclass="pre">np.array</span></code>, and resolve the two dimensions of data, which we deem the ‘student’ dimension and the ‘exam’ dimension, respectively.</p>
<divclass="admonition warning">
<pclass="admonition-title fa fa-exclamation-circle"><strong>Axis vs Dimension</strong>:</p>
<p>Although NumPy does formally recognize the concept of dimensionality precisely in the way that it is discussed here, its documentation refers to an individual dimension of an array as an <strong>axis</strong>. Thus you will see “axes” (pronounced “aks-ēz”) used in place of “dimensions”; however, they mean the same thing.</p>
</div>
<p>NumPy specifies the row-axis (students) of a 2D array as “axis-0” and the column-axis (exams) as axis-1. You must now provide <em>two</em> indices, one for each axis (dimension), to uniquely specify an element in this 2D array; the first number specifies an index along axis-0, the second specifies an index along axis-1. The zero-based indexing schema that we reviewed earlier applies to each axis of the ND-array:</p>
<divclass="highlight-none notranslate"><divclass="highlight"><pre><span></span> -- axis-1 ->
-2 -1
0 1
| +---+---+
| -3, 0 |93 | 95|
| +---+---+
axis-0 -2, 1 |84 |100|
| +---+---+
| -1, 2 |99 | 87|
V +---+---+
</pre></div>
</div>
<p>Because <codeclass="docutils literal notranslate"><spanclass="pre">grades</span></code> has three entries along axis-0 and two entries along axis-1, it has a “shape” of <codeclass="docutils literal notranslate"><spanclass="pre">(3,</span><spanclass="pre">2)</span></code>.</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="gp">>>> </span><spanclass="n">grades</span><spanclass="o">.</span><spanclass="n">shape</span>
<spanclass="go">(3, 2)</span>
</pre></div>
</div>
<divclass="section" id="Integer-Indexing">
<h3>Integer Indexing<aclass="headerlink" href="#Integer-Indexing" title="Permalink to this headline"></a></h3>
<p>Thus, if we want to access Brad’s (item-1 along axis-0) score for Exam 1 (item-0 along axis-1) we simply specify:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># providing two numbers to access an element</span>
<spanclass="c1"># in a 2D-array</span>
<spanclass="o">>>></span><spanclass="n">grades</span><spanclass="p">[</span><spanclass="mi">1</span><spanclass="p">,</span><spanclass="mi">0</span><spanclass="p">]</span><spanclass="c1"># Brad's score on Exam 1</span>
<spanclass="mi">84</span>
<spanclass="c1"># negative indices work as with lists/tuples/strings</span>
<spanclass="o">>>></span><spanclass="n">grades</span><spanclass="p">[</span><spanclass="o">-</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">0</span><spanclass="p">]</span><spanclass="c1"># Brad's score on Exam 1</span>
<spanclass="mi">84</span>
</pre></div>
</div>
</div>
<divclass="section" id="Slice-Indexing">
<h3>Slice Indexing<aclass="headerlink" href="#Slice-Indexing" title="Permalink to this headline"></a></h3>
<p>We can also uses <em>slices</em> to access subsequences of our data. Suppose we want the scores of all the students for Exam 2. We can slice from 0 through 3 along axis-0 (refer to the indexing diagram in the previous section) to include all the students, and specify index 1 on axis-1 to select Exam 2:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="gp">>>> </span><spanclass="n">grades</span><spanclass="p">[</span><spanclass="mi">0</span><spanclass="p">:</span><spanclass="mi">3</span><spanclass="p">,</span><spanclass="mi">1</span><spanclass="p">]</span><spanclass="c1"># Exam 2 scores for all students</span>
<spanclass="go">array([ 95, 100, 87])</span>
</pre></div>
</div>
<p>As with Python sequences, you can specify an “empty” slice to include all possible entries along an axis, by default: <codeclass="docutils literal notranslate"><spanclass="pre">grades[:,</span><spanclass="pre">1]</span></code> is equivalent to <codeclass="docutils literal notranslate"><spanclass="pre">grades[0:3,</span><spanclass="pre">1]</span></code>, in this instance. More generally, withholding either the ‘start’ or ‘stop’ value in a slice will result in the use smallest or largest valid index, respectively:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="gp">>>> </span><spanclass="n">grades</span><spanclass="p">[</span><spanclass="mi">1</span><spanclass="p">:,</span><spanclass="mi">1</span><spanclass="p">]</span><spanclass="c1"># equivalent to `grades[1:3, 1]</span>
<spanclass="go">array([ 100, 87])</span>
<spanclass="gp">>>> </span><spanclass="n">grades</span><spanclass="p">[:,</span><spanclass="p">:</span><spanclass="mi">1</span><spanclass="p">]</span><spanclass="c1"># equivalent to `grades[0:3, 0:1]</span>
<spanclass="go">array([[93],</span>
<spanclass="go"> [84],</span>
<spanclass="go"> [99]])</span>
</pre></div>
</div>
<p>The output of <codeclass="docutils literal notranslate"><spanclass="pre">grades[:,</span><spanclass="pre">:1]</span></code> might look somewhat funny. Because the axis-1 slice only includes one column of numbers, the shape of the resulting array is (3, 1). 0 is thus only valid (non-negative) index for axis-1, since there is only one column to specify in the array.</p>
<p>You can also supply a “step” value to the slice. <codeclass="docutils literal notranslate"><spanclass="pre">grades[::-1,</span><spanclass="pre">:]</span></code> will returns the array of grades with the student-axis flipped (reverse-alphabetical order).</p>
</div>
<divclass="section" id="Negative-Indices">
<h3>Negative Indices<aclass="headerlink" href="#Negative-Indices" title="Permalink to this headline"></a></h3>
<p>As indicated above, negative indices are valid too and are quite useful. If we want to access the scores of the latest exam for all of the students, you can specify:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># using a negative index and a slice</span>
<spanclass="o">>>></span><spanclass="n">grades</span><spanclass="p">[:,</span><spanclass="o">-</span><spanclass="mi">1</span><spanclass="p">]</span><spanclass="c1"># Latest exam scores (Exam 2), for all students</span>
<spanclass="n">array</span><spanclass="p">([</span><spanclass="mi">95</span><spanclass="p">,</span><spanclass="mi">100</span><spanclass="p">,</span><spanclass="mi">87</span><spanclass="p">])</span>
</pre></div>
</div>
<p>Note the value of using the negative index is that it will always provide you with the latest exam score - you need not check how many exams the students have taken.</p>
</div>
<divclass="section" id="Supplying-Fewer-Indices-Than-Dimensions">
<h3>Supplying Fewer Indices Than Dimensions<aclass="headerlink" href="#Supplying-Fewer-Indices-Than-Dimensions" title="Permalink to this headline"></a></h3>
<p>What happens if we only supply one index to our array? It may be surprising that <codeclass="docutils literal notranslate"><spanclass="pre">grades[0]</span></code> does not throw an error since we are specifying only one index to access data from a 2-dimensional array. Instead, NumPy it will return all of the exam scores for student-0 (Ashley):</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="gp">>>> </span><spanclass="n">grades</span><spanclass="p">[</span><spanclass="mi">0</span><spanclass="p">]</span>
<spanclass="go">array([ 93, 95])</span>
</pre></div>
</div>
<p>This is because NumPy will automatically insert trailing slices for you if you don’t provide as many indices as there are dimensions for your array. <codeclass="docutils literal notranslate"><spanclass="pre">grades[0]</span></code> was treated as <codeclass="docutils literal notranslate"><spanclass="pre">grades[0,</span><spanclass="pre">:]</span></code>.</p>
<divclass="admonition note">
<p>Suppose you have an <spanclass="math notranslate nohighlight">\(N\)</span>-dimensional array, and only provide <spanclass="math notranslate nohighlight">\(j\)</span> indices for the array; NumPy will automatically insert <spanclass="math notranslate nohighlight">\(N-j\)</span> trailing slices for you. In the case that <spanclass="math notranslate nohighlight">\(N=5\)</span> and <spanclass="math notranslate nohighlight">\(j=3\)</span>, <codeclass="docutils literal notranslate"><spanclass="pre">d5_array[0,</span><spanclass="pre">0,</span><spanclass="pre">0]</span></code> is treated as <codeclass="docutils literal notranslate"><spanclass="pre">d5_array[0,</span><spanclass="pre">0,</span><spanclass="pre">0,</span><spanclass="pre">:,</span><spanclass="pre">:]</span></code></p>
</div>
<p>Thus far, we have discussed some rules for accessing data in arrays, all of which fall into the category that is designated <aclass="reference external" href="https://numpy.org/doc/stable/reference/arrays.indexing.html#basic-slicing-and-indexing">“basic indexing”</a> by the NumPy documentation. We will discuss the details of basic indexing and of <aclass="reference external" href="https://numpy.org/doc/stable/reference/arrays.indexing.html#advanced-indexing">“advanced indexing”</a>, in full, in a later section. Note, however, that all of the indexing/slicing
reviewed here produces a “view” of the original array. That is, <em>no data is copied</em> when you index into an array using integer indices and/or slices. Recall that slicing lists and tuples <em>do</em> produce copies of the data.</p>
<divclass="admonition warning">
<pclass="admonition-title fa fa-exclamation-circle"><strong>FYI</strong>:</p>
<p>Keeping track of the meaning of an array’s various dimensions can quickly become unwieldy when working with real datasets. <aclass="reference external" href="http://xarray.pydata.org/en/stable/">xarray</a> is a Python library that provides functionality comparable to NumPy, but allows users provide <em>explicit labels</em> for an array’s dimensions; that is, you can <em>name</em> each dimension. Using an <codeclass="docutils literal notranslate"><spanclass="pre">xarray</span></code> to select Brad’s scores could look like <codeclass="docutils literal notranslate"><spanclass="pre">grades.sel(student='Brad')</span></code>, for instance. This is a valuable library to look into at
your leisure.</p>
</div>
</div>
</div>
<divclass="section" id="N-dimensional-Arrays">
<h2>N-dimensional Arrays<aclass="headerlink" href="#N-dimensional-Arrays" title="Permalink to this headline"></a></h2>
<p>Let’s build up some intuition for arrays with a dimensionality higher than 2. The following code creates a 3-dimensional array:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># a 3D array, shape-(2, 2, 2)</span>
<spanclass="o">>>></span><spanclass="n">d3_array</span><spanclass="o">=</span><spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">([[[</span><spanclass="mi">0</span><spanclass="p">,</span><spanclass="mi">1</span><spanclass="p">],</span>
<spanclass="o">...</span><spanclass="p">[</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">3</span><spanclass="p">]],</span>
<spanclass="o">...</span>
<spanclass="o">...</span><spanclass="p">[[</span><spanclass="mi">4</span><spanclass="p">,</span><spanclass="mi">5</span><spanclass="p">],</span>
<spanclass="o">...</span><spanclass="p">[</span><spanclass="mi">6</span><spanclass="p">,</span><spanclass="mi">7</span><spanclass="p">]]])</span>
</pre></div>
</div>
<p>You can think of axis-0 denoting which of the 2x2 “sheets” to select from. Then axis-1 specifies the row along the sheets, and axis-2 the column within the row:</p>
<p><strong>Depicting the layout of a 3D array</strong></p>
<divclass="highlight-none notranslate"><divclass="highlight"><pre><span></span>sheet 0:
[0, 1]
[2, 3]
sheet 1:
[4, 5]
[6, 7]
</pre></div>
</div>
<divclass="highlight-none notranslate"><divclass="highlight"><pre><span></span> | -- axis-2 ->
| |
| axis-1 [0, 1]
| | [2, 3]
| V
axis-0
| -- axis-2 ->
| |
| axis-1 [4, 5]
| | [6, 7]
V V
</pre></div>
</div>
<p>Thus <codeclass="docutils literal notranslate"><spanclass="pre">d3_array[0,</span><spanclass="pre">1,</span><spanclass="pre">0]</span></code> specifies the element residing in sheet-0, at row-1 and column-0:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># retrieving a single element from a 3D-array</span>
<spanclass="o">>>></span><spanclass="n">d3_array</span><spanclass="p">[</span><spanclass="mi">0</span><spanclass="p">,</span><spanclass="mi">1</span><spanclass="p">,</span><spanclass="mi">0</span><spanclass="p">]</span>
<spanclass="mi">2</span>
</pre></div>
</div>
<p><codeclass="docutils literal notranslate"><spanclass="pre">d3_array[:,</span><spanclass="pre">0,</span><spanclass="pre">0]</span></code> specifies the elements in row-0 and column-0 of <strong>both</strong> sheets:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># retrieving a 1D sub-array from a 3D-array</span>
<spanclass="o">>>></span><spanclass="n">d3_array</span><spanclass="p">[:,</span><spanclass="mi">0</span><spanclass="p">,</span><spanclass="mi">0</span><spanclass="p">]</span>
<spanclass="n">array</span><spanclass="p">([</span><spanclass="mi">0</span><spanclass="p">,</span><spanclass="mi">4</span><spanclass="p">])</span>
</pre></div>
</div>
<p><codeclass="docutils literal notranslate"><spanclass="pre">d3_array[1]</span></code>, which recall is shorthand for <codeclass="docutils literal notranslate"><spanclass="pre">d3_array[1,</span><spanclass="pre">:,</span><spanclass="pre">:]</span></code>, selects both rows and both columns of sheet-1:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># retrieving a 2D sub-array from a 3D-array</span>
<spanclass="o">>>></span><spanclass="n">d3_array</span><spanclass="p">[</span><spanclass="mi">1</span><spanclass="p">]</span>
<spanclass="n">array</span><spanclass="p">([[</span><spanclass="mi">4</span><spanclass="p">,</span><spanclass="mi">5</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">6</span><spanclass="p">,</span><spanclass="mi">7</span><spanclass="p">]])</span>
</pre></div>
</div>
<p>In four dimensions, one can think of “<em>stacks</em> of sheets with rows and columns” where axis-0 selects the stack of sheets you are working with, axis-1 chooses the sheet, axis-2 chooses the row, and axis-3 chooses the column. Extrapolating to higher dimensions (“collections of stacks of sheets …”) continues in the same tedious fashion.</p>
<divclass="admonition note">
<pclass="admonition-title fa fa-exclamation-circle"><strong>Reading Comprehension: Multi-dimensional Indexing</strong></p>
<p>Given the 3D, shape-(3, 3, 3) array:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="gp">>>> </span><spanclass="n">arr</span><spanclass="o">=</span><spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">([[[</span><spanclass="mi">0</span><spanclass="p">,</span><spanclass="mi">1</span><spanclass="p">,</span><spanclass="mi">2</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">3</span><spanclass="p">,</span><spanclass="mi">4</span><spanclass="p">,</span><spanclass="mi">5</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">6</span><spanclass="p">,</span><spanclass="mi">7</span><spanclass="p">,</span><spanclass="mi">8</span><spanclass="p">]],</span>
<spanclass="gp">...</span>
<spanclass="gp">... </span><spanclass="p">[[</span><spanclass="mi">9</span><spanclass="p">,</span><spanclass="mi">10</span><spanclass="p">,</span><spanclass="mi">11</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">12</span><spanclass="p">,</span><spanclass="mi">13</span><spanclass="p">,</span><spanclass="mi">14</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">15</span><spanclass="p">,</span><spanclass="mi">16</span><spanclass="p">,</span><spanclass="mi">17</span><spanclass="p">]],</span>
<spanclass="gp">...</span>
<spanclass="gp">... </span><spanclass="p">[[</span><spanclass="mi">18</span><spanclass="p">,</span><spanclass="mi">19</span><spanclass="p">,</span><spanclass="mi">20</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">21</span><spanclass="p">,</span><spanclass="mi">22</span><spanclass="p">,</span><spanclass="mi">23</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">24</span><spanclass="p">,</span><spanclass="mi">25</span><spanclass="p">,</span><spanclass="mi">26</span><spanclass="p">]]])</span>
</pre></div>
</div>
<p>Index into the array to produce the following results</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1">#1</span>
<spanclass="n">array</span><spanclass="p">([[</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">5</span><spanclass="p">,</span><spanclass="mi">8</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">11</span><spanclass="p">,</span><spanclass="mi">14</span><spanclass="p">,</span><spanclass="mi">17</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">20</span><spanclass="p">,</span><spanclass="mi">23</span><spanclass="p">,</span><spanclass="mi">26</span><spanclass="p">]])</span>
<spanclass="c1">#2</span>
<spanclass="n">array</span><spanclass="p">([[</span><spanclass="mi">3</span><spanclass="p">,</span><spanclass="mi">4</span><spanclass="p">,</span><spanclass="mi">5</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">12</span><spanclass="p">,</span><spanclass="mi">13</span><spanclass="p">,</span><spanclass="mi">14</span><spanclass="p">]])</span>
<spanclass="c1">#3</span>
<spanclass="n">array</span><spanclass="p">([</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">5</span><spanclass="p">])</span>
<spanclass="c1">#4</span>
<spanclass="n">array</span><spanclass="p">([[</span><spanclass="mi">11</span><spanclass="p">,</span><spanclass="mi">10</span><spanclass="p">,</span><spanclass="mi">9</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">14</span><spanclass="p">,</span><spanclass="mi">13</span><spanclass="p">,</span><spanclass="mi">12</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">17</span><spanclass="p">,</span><spanclass="mi">16</span><spanclass="p">,</span><spanclass="mi">15</span><spanclass="p">]])</span>
</pre></div>
</div>
</div>
</div>
<divclass="section" id="Zero-dimensional-Arrays">
<h2>Zero-dimensional Arrays<aclass="headerlink" href="#Zero-dimensional-Arrays" title="Permalink to this headline"></a></h2>
<p>A zero dimensional array is simply a single number (a.k.a. a scalar value):</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># creating a 0-dimensional array</span>
<spanclass="o">>>></span><spanclass="n">x</span><spanclass="o">=</span><spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">(</span><spanclass="mf">15.2</span><spanclass="p">)</span>
</pre></div>
</div>
<p>This is <em>not</em> equivalent to a length-1 1D-array: <codeclass="docutils literal notranslate"><spanclass="pre">np.array([15.2])</span></code>. According to our definition of dimensionality, <em>zero</em> numbers are required to index into a 0-D array as it is unnecessary to provide an identifier for a standalone number. Thus you cannot index into a 0-D array.</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># you cannot index into a 0-D array</span>
<spanclass="o">>>></span><spanclass="n">x</span><spanclass="p">[</span><spanclass="mi">0</span><spanclass="p">]</span>
<spanclass="o">---------------------------------------------------------------------------</span>
<spanclass="ne">IndexError</span><spanclass="n">Traceback</span><spanclass="p">(</span><spanclass="n">most</span><spanclass="n">recent</span><spanclass="n">call</span><spanclass="n">last</span><spanclass="p">)</span>
<spanclass="o"><</span><spanclass="n">ipython</span><spanclass="o">-</span><spanclass="nb">input</span><spanclass="o">-</span><spanclass="mi">10</span><spanclass="o">-</span><spanclass="mi">2</span><spanclass="n">f755f117ac9</span><spanclass="o">></span><spanclass="ow">in</span><spanclass="o"><</span><spanclass="n">module</span><spanclass="o">></span><spanclass="p">()</span>
<spanclass="o">----></span><spanclass="mi">1</span><spanclass="n">x</span><spanclass="p">[</span><spanclass="mi">0</span><spanclass="p">]</span>
<spanclass="ne">IndexError</span><spanclass="p">:</span><spanclass="n">too</span><spanclass="n">many</span><spanclass="n">indices</span><spanclass="k">for</span><spanclass="n">array</span>
</pre></div>
</div>
<p>You must use the syntax <codeclass="docutils literal notranslate"><spanclass="pre">arr.item()</span></code> to retrieve the numerical entry from a 0D array:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="gp">>>> </span><spanclass="n">x</span><spanclass="o">.</span><spanclass="n">item</span><spanclass="p">()</span>
<spanclass="go">15.2</span>
</pre></div>
</div>
<p>Zero-dimensional arrays do not show up in real applications very often. They are, however, important from the point of view of NumPy being self-consistent in how it treats dimensionality in its arrays, and it is important that you are at least exposed to a 0D array and understand its nuances.</p>
<divclass="admonition note">
<pclass="admonition-title fa fa-exclamation-circle"><strong>Takeaway</strong>:</p>
<p>Although accessing data along varying dimensions is ultimately all a matter of judicious bookkeeping (you <em>could</em> access all of this data from a 1-dimensional array, after all), NumPy’s ability to provide users with an interface for accessing data along dimensions is incredibly useful. It affords us an ability to impose intuitive, abstract structure to our data.</p>
</div>
</div>
<divclass="section" id="Manipulating-Arrays">
<h2>Manipulating Arrays<aclass="headerlink" href="#Manipulating-Arrays" title="Permalink to this headline"></a></h2>
<p>NumPy provides an assortment of functions that allow us manipulate the way that an array’s data can be accessed. These permit us to reshape an array, change its dimensionality, and swap the positions of its axes:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="gp">>>> </span><spanclass="n">x</span><spanclass="o">=</span><spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">([[</span><spanclass="mi">1</span><spanclass="p">,</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">3</span><spanclass="p">,</span><spanclass="mi">4</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">5</span><spanclass="p">,</span><spanclass="mi">6</span><spanclass="p">,</span><spanclass="mi">7</span><spanclass="p">,</span><spanclass="mi">8</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">9</span><spanclass="p">,</span><spanclass="mi">10</span><spanclass="p">,</span><spanclass="mi">11</span><spanclass="p">,</span><spanclass="mi">12</span><spanclass="p">]])</span>
<spanclass="go"># reshaping an array</span>
<spanclass="gp">>>> </span><spanclass="n">x</span><spanclass="o">.</span><spanclass="n">reshape</span><spanclass="p">(</span><spanclass="mi">3</span><spanclass="p">,</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">2</span><spanclass="p">)</span>
<spanclass="go">array([[[ 1, 2],</span>
<spanclass="go"> [ 3, 4]],</span>
<spanclass="go"> [[ 5, 6],</span>
<spanclass="go"> [ 7, 8]],</span>
<spanclass="go"> [[ 9, 10],</span>
<spanclass="go"> [11, 12]]])</span>
<spanclass="go"># Transposing an array: reversing</span>
<spanclass="go"># the ordering of its axes. This interchanges</span>
<spanclass="go"># the rows and columns of `x`</span>
<spanclass="gp">>>> </span><spanclass="n">x</span><spanclass="o">.</span><spanclass="n">transpose</span><spanclass="p">()</span>
<spanclass="go">array([[ 1, 5, 9],</span>
<spanclass="go"> [ 2, 6, 10],</span>
<spanclass="go"> [ 3, 7, 11],</span>
<spanclass="go"> [ 4, 8, 12]])</span>
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</div>
<p>A complete listing of the available array-manipulation functions can be found in the <aclass="reference external" href="https://numpy.org/doc/stable/reference/routines.array-manipulation.html">official NumPy documentation</a>. Among these functions, the reshape function is especially useful.</p>
<divclass="section" id="Introducing-the-reshape-Function">
<h3>Introducing the <codeclass="docutils literal notranslate"><spanclass="pre">reshape</span></code> Function<aclass="headerlink" href="#Introducing-the-reshape-Function" title="Permalink to this headline"></a></h3>
<p>The <codeclass="docutils literal notranslate"><spanclass="pre">reshape</span></code> function allows you to change the dimensionality and axis-layout of a given array. This adjusts the indexing interface used to access the array’s underlying data, as was discussed in earlier in this module. Let’s take a shape-(6,) array, and reshape it to a shape-(2, 3) array:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="gp">>>> </span><spanclass="kn">import</span><spanclass="nn">numpy</span><spanclass="k">as</span><spanclass="nn">np</span>
<spanclass="gp">>>> </span><spanclass="n">x</span><spanclass="o">=</span><spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">([</span><spanclass="mi">0</span><spanclass="p">,</span><spanclass="mi">1</span><spanclass="p">,</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">3</span><spanclass="p">,</span><spanclass="mi">4</span><spanclass="p">,</span><spanclass="mi">5</span><spanclass="p">])</span>
<spanclass="go"># reshape a shape-(6,) array into a shape-(2,3) array</span>
<spanclass="gp">>>> </span><spanclass="n">x</span><spanclass="o">.</span><spanclass="n">reshape</span><spanclass="p">(</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">3</span><spanclass="p">)</span>
<spanclass="go">array([[0, 1, 2],</span>
<spanclass="go"> [3, 4, 5]])</span>
</pre></div>
</div>
<p>You can also conveniently reshape an array by “setting” its shape via assignment:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># equivalent to: x = x.reshape(2, 3)</span>
<spanclass="o">>>></span><spanclass="n">x</span><spanclass="o">.</span><spanclass="n">shape</span><spanclass="o">=</span><spanclass="p">(</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">3</span><spanclass="p">)</span>
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</div>
<p>Of course, the size the the initial array must match the size of the to-be reshaped array:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># an array with 5 numbers are cannot be reshaped</span>
<spanclass="c1"># into a (3, 2) array</span>
<spanclass="o">>>></span><spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">([</span><spanclass="mi">0</span><spanclass="p">,</span><spanclass="mi">1</span><spanclass="p">,</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">3</span><spanclass="p">,</span><spanclass="mi">4</span><spanclass="p">])</span><spanclass="o">.</span><spanclass="n">reshape</span><spanclass="p">(</span><spanclass="mi">3</span><spanclass="p">,</span><spanclass="mi">2</span><spanclass="p">)</span>
<spanclass="ne">ValueError</span><spanclass="p">:</span><spanclass="n">total</span><spanclass="n">size</span><spanclass="n">of</span><spanclass="n">new</span><spanclass="n">array</span><spanclass="n">must</span><spanclass="n">be</span><spanclass="n">unchanged</span>
</pre></div>
</div>
<p>Multidimensional arrays can be reshaped too:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># reshaping a multidimensional array</span>
<spanclass="o">>>></span><spanclass="n">x</span><spanclass="o">=</span><spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">([[</span><spanclass="mi">0</span><spanclass="p">,</span><spanclass="mi">1</span><spanclass="p">,</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">3</span><spanclass="p">],</span>
<spanclass="o">...</span><spanclass="p">[</span><spanclass="mi">4</span><spanclass="p">,</span><spanclass="mi">5</span><spanclass="p">,</span><spanclass="mi">6</span><spanclass="p">,</span><spanclass="mi">7</span><spanclass="p">],</span>
<spanclass="o">...</span><spanclass="p">[</span><spanclass="mi">8</span><spanclass="p">,</span><spanclass="mi">9</span><spanclass="p">,</span><spanclass="mi">10</span><spanclass="p">,</span><spanclass="mi">11</span><spanclass="p">]])</span>
<spanclass="c1"># reshape from (3, 4) to (2, 3, 2)</span>
<spanclass="o">>>></span><spanclass="n">x</span><spanclass="o">.</span><spanclass="n">reshape</span><spanclass="p">(</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">3</span><spanclass="p">,</span><spanclass="mi">2</span><spanclass="p">)</span>
<spanclass="n">array</span><spanclass="p">([[[</span><spanclass="mi">0</span><spanclass="p">,</span><spanclass="mi">1</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">3</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">4</span><spanclass="p">,</span><spanclass="mi">5</span><spanclass="p">]],</span>
<spanclass="p">[[</span><spanclass="mi">6</span><spanclass="p">,</span><spanclass="mi">7</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">8</span><spanclass="p">,</span><spanclass="mi">9</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">10</span><spanclass="p">,</span><spanclass="mi">11</span><spanclass="p">]]])</span>
</pre></div>
</div>
<p>Because the size of an input array and the resulting reshaped array must agree, you can specify <em>one</em> of the dimension-sizes in the reshape function to be -1, and this will cue NumPy to compute that dimension’s size for you. For example, if you are reshaping a shape-(36,) array into a shape-(3, 4, 3) array. The following are valid:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># Equivalent ways of specifying a reshape</span>
<spanclass="c1"># np.arange(36) produces the shape-(36,) array ([0, 1, 2, ..., 35])</span>
<spanclass="n">np</span><spanclass="o">.</span><spanclass="n">arange</span><spanclass="p">(</span><spanclass="mi">36</span><spanclass="p">)</span><spanclass="o">.</span><spanclass="n">reshape</span><spanclass="p">(</span><spanclass="mi">3</span><spanclass="p">,</span><spanclass="mi">4</span><spanclass="p">,</span><spanclass="mi">3</span><spanclass="p">)</span><spanclass="c1"># (36,) --reshape--> (3, 4, 3)</span>
<spanclass="n">np</span><spanclass="o">.</span><spanclass="n">arange</span><spanclass="p">(</span><spanclass="mi">36</span><spanclass="p">)</span><spanclass="o">.</span><spanclass="n">reshape</span><spanclass="p">(</span><spanclass="mi">3</span><spanclass="p">,</span><spanclass="mi">4</span><spanclass="p">,</span><spanclass="o">-</span><spanclass="mi">1</span><spanclass="p">)</span><spanclass="c1"># NumPy replaces -1 with 36/(3*4) -> 3</span>
<spanclass="n">np</span><spanclass="o">.</span><spanclass="n">arange</span><spanclass="p">(</span><spanclass="mi">36</span><spanclass="p">)</span><spanclass="o">.</span><spanclass="n">reshape</span><spanclass="p">(</span><spanclass="mi">3</span><spanclass="p">,</span><spanclass="o">-</span><spanclass="mi">1</span><spanclass="p">,</span><spanclass="mi">3</span><spanclass="p">)</span><spanclass="c1"># NumPy replaces -1 with 36/(3*3) -> 4</span>
<spanclass="n">np</span><spanclass="o">.</span><spanclass="n">arange</span><spanclass="p">(</span><spanclass="mi">36</span><spanclass="p">)</span><spanclass="o">.</span><spanclass="n">reshape</span><spanclass="p">(</span><spanclass="o">-</span><spanclass="mi">1</span><spanclass="p">,</span><spanclass="mi">4</span><spanclass="p">,</span><spanclass="mi">3</span><spanclass="p">)</span><spanclass="c1"># NumPy replaces -1 with 36/(3*4) -> 3</span>
</pre></div>
</div>
<p>You can use -1 to specify only one dimension:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="gp">>>> </span><spanclass="n">np</span><spanclass="o">.</span><spanclass="n">arange</span><spanclass="p">(</span><spanclass="mi">36</span><spanclass="p">)</span><spanclass="o">.</span><spanclass="n">reshape</span><spanclass="p">(</span><spanclass="mi">3</span><spanclass="p">,</span><spanclass="o">-</span><spanclass="mi">1</span><spanclass="p">,</span><spanclass="o">-</span><spanclass="mi">1</span><spanclass="p">)</span><spanclass="c1"># this is an ambiguous specification, and thus</span>
<spanclass="go">---------------------------------------------------------------------------</span>
<spanclass="go">ValueError Traceback (most recent call last)</span>
<spanclass="go"><ipython-input-3-207d18d18af2> in <module>()</span>
<spanclass="go">----> 1 np.arange(36).reshape(3, -1, -1)</span>
<spanclass="go">ValueError: can only specify one unknown dimension</span>
</pre></div>
</div>
<divclass="admonition note">
<pclass="admonition-title fa fa-exclamation-circle"><strong>Reshaping Does Not Make a Copy of an Array</strong>:</p>
<p>For all straightforward applications of reshape, NumPy does not actually create a new copy of an array’s data when performing a <codeclass="docutils literal notranslate"><spanclass="pre">reshape</span></code> operation. Instead, the original array and the reshaped array reference the same underlying data. The reshaped array simply provides a new index-interface for accessing said data, and is thus referred to as a “view” of the original array (more on this “views” in a later section).</p>
</div>
</div>
</div>
<divclass="section" id="Links-to-Official-Documentation">
<h2>Links to Official Documentation<aclass="headerlink" href="#Links-to-Official-Documentation" title="Permalink to this headline"></a></h2>
<ulclass="simple">
<li><p><aclass="reference external" href="https://numpy.org/doc/stable/reference/arrays.ndarray.html">The N-dimensional array</a></p></li>
<li><p><aclass="reference external" href="https://numpy.org/doc/stable/user/basics.indexing.html#indexing">Array indexing</a></p></li>
<li><p><aclass="reference external" href="https://numpy.org/doc/stable/reference/routines.indexing.html#indexing-routines">Indexing routines</a></p></li>
<li><p><aclass="reference external" href="https://numpy.org/doc/stable/reference/routines.array-manipulation.html">Array manipulation routines</a></p></li>
</ul>
</div>
<divclass="section" id="Reading-Comprehension-Solutions">
<h2>Reading Comprehension Solutions<aclass="headerlink" href="#Reading-Comprehension-Solutions" title="Permalink to this headline"></a></h2>
<p><strong>Reading Comprehension: Multi-dimensional Indexing</strong></p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="gp">>>> </span><spanclass="n">arr</span><spanclass="o">=</span><spanclass="n">np</span><spanclass="o">.</span><spanclass="n">array</span><spanclass="p">([[[</span><spanclass="mi">0</span><spanclass="p">,</span><spanclass="mi">1</span><spanclass="p">,</span><spanclass="mi">2</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">3</span><spanclass="p">,</span><spanclass="mi">4</span><spanclass="p">,</span><spanclass="mi">5</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">6</span><spanclass="p">,</span><spanclass="mi">7</span><spanclass="p">,</span><spanclass="mi">8</span><spanclass="p">]],</span>
<spanclass="gp">...</span>
<spanclass="gp">... </span><spanclass="p">[[</span><spanclass="mi">9</span><spanclass="p">,</span><spanclass="mi">10</span><spanclass="p">,</span><spanclass="mi">11</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">12</span><spanclass="p">,</span><spanclass="mi">13</span><spanclass="p">,</span><spanclass="mi">14</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">15</span><spanclass="p">,</span><spanclass="mi">16</span><spanclass="p">,</span><spanclass="mi">17</span><spanclass="p">]],</span>
<spanclass="gp">...</span>
<spanclass="gp">... </span><spanclass="p">[[</span><spanclass="mi">18</span><spanclass="p">,</span><spanclass="mi">19</span><spanclass="p">,</span><spanclass="mi">20</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">21</span><spanclass="p">,</span><spanclass="mi">22</span><spanclass="p">,</span><spanclass="mi">23</span><spanclass="p">],</span>
<spanclass="gp">... </span><spanclass="p">[</span><spanclass="mi">24</span><spanclass="p">,</span><spanclass="mi">25</span><spanclass="p">,</span><spanclass="mi">26</span><spanclass="p">]]])</span>
</pre></div>
</div>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1">#1</span>
<spanclass="o">>>></span><spanclass="n">arr</span><spanclass="p">[:,</span><spanclass="p">:,</span><spanclass="mi">2</span><spanclass="p">]</span>
<spanclass="n">array</span><spanclass="p">([[</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">5</span><spanclass="p">,</span><spanclass="mi">8</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">11</span><spanclass="p">,</span><spanclass="mi">14</span><spanclass="p">,</span><spanclass="mi">17</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">20</span><spanclass="p">,</span><spanclass="mi">23</span><spanclass="p">,</span><spanclass="mi">26</span><spanclass="p">]])</span>
<spanclass="c1">#2</span>
<spanclass="o">>>></span><spanclass="n">arr</span><spanclass="p">[</span><spanclass="mi">0</span><spanclass="p">:</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">1</span><spanclass="p">,</span><spanclass="p">:]</span>
<spanclass="n">array</span><spanclass="p">([[</span><spanclass="mi">3</span><spanclass="p">,</span><spanclass="mi">4</span><spanclass="p">,</span><spanclass="mi">5</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">12</span><spanclass="p">,</span><spanclass="mi">13</span><spanclass="p">,</span><spanclass="mi">14</span><spanclass="p">]])</span>
<spanclass="c1">#3</span>
<spanclass="o">>>></span><spanclass="n">arr</span><spanclass="p">[</span><spanclass="mi">0</span><spanclass="p">,</span><spanclass="p">:</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">2</span><spanclass="p">]</span>
<spanclass="n">array</span><spanclass="p">([</span><spanclass="mi">2</span><spanclass="p">,</span><spanclass="mi">5</span><spanclass="p">])</span>
<spanclass="c1">#4</span>
<spanclass="o">>>></span><spanclass="n">arr</span><spanclass="p">[</span><spanclass="mi">1</span><spanclass="p">,</span><spanclass="p">:,</span><spanclass="p">::</span><spanclass="o">-</span><spanclass="mi">1</span><spanclass="p">]</span>
<spanclass="n">array</span><spanclass="p">([[</span><spanclass="mi">11</span><spanclass="p">,</span><spanclass="mi">10</span><spanclass="p">,</span><spanclass="mi">9</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">14</span><spanclass="p">,</span><spanclass="mi">13</span><spanclass="p">,</span><spanclass="mi">12</span><spanclass="p">],</span>
<spanclass="p">[</span><spanclass="mi">17</span><spanclass="p">,</span><spanclass="mi">16</span><spanclass="p">,</span><spanclass="mi">15</span><spanclass="p">]])</span>
</pre></div>
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