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Practice and theory of brain imaging
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">The neuroimaging problem</span></p>
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<li class="toctree-l1"><a class="reference internal" href="the_problem.html">The problem and the plan</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">The tools</span></p>
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<li class="toctree-l1"><a class="reference internal" href="to_code.html">Ode to code</a></li>
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<li class="toctree-l1"><a class="reference internal" href="using_jupyter.html">Using Jupyter notebooks</a></li>
<li class="toctree-l1"><a class="reference internal" href="more_on_jupyter.html">More on the Jupyter notebook</a></li>
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<li class="toctree-l1"><a class="reference internal" href="installation.html">Installing on your computer</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Images and code and images</span></p>
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<li class="toctree-l1"><a class="reference internal" href="pathlib.html">Using the <code class="docutils literal notranslate"><span class="pre">pathlib</span></code> module</a></li>
<li class="toctree-l1"><a class="reference internal" href="what_is_an_image.html">What is an image?</a></li>
<li class="toctree-l1"><a class="reference internal" href="numpy_intro.html">Introduction to Numpy</a></li>
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<li class="toctree-l1"><a class="reference internal" href="reshape_and_4d.html">Reshaping, 4D to 2D</a></li>
<li class="toctree-l1"><a class="reference internal" href="voxels_by_time.html">Voxels and time</a></li>
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<li class="toctree-l1"><a class="reference internal" href="assert.html">Using <code class="docutils literal notranslate"><span class="pre">assert</span></code> for testing</a></li>
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<li class="toctree-l1"><a class="reference internal" href="on_testing.html">On testing</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Detecting activation</span></p>
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<li class="toctree-l1"><a class="reference internal" href="voxel_time_courses.html">Correlating with voxel time courses</a></li>
<li class="toctree-l1"><a class="reference internal" href="on_convolution.html">Convolution</a></li>
<li class="toctree-l1"><a class="reference internal" href="convolution_background.html">Convolving with the hemodyamic response function</a></li>
<li class="toctree-l1"><a class="reference internal" href="on_regression.html">Introducing regression</a></li>
<li class="toctree-l1"><a class="reference internal" href="regress_one_voxel.html">Regression for a single voxel</a></li>
<li class="toctree-l1"><a class="reference internal" href="regression_notation.html">Notation for regression models</a></li>
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<li class="toctree-l1"><a class="reference internal" href="test_one_voxel.html">Testing a single voxel</a></li>
<li class="toctree-l1"><a class="reference internal" href="non_tr_onsets.html">Using onsets that do not start on a TR</a></li>
<li class="toctree-l1"><a class="reference internal" href="whole_image_statistics.html">Whole-brain t tests and p values</a></li>
<li class="toctree-l1"><a class="reference internal" href="bonferroni_correction.html">Notes on the Bonferroni threshold</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Spatial processing</span></p>
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<li class="toctree-l1"><a class="reference internal" href="diagonal_zooms.html">Encoding zooms (scaling) with a diagonal matrix</a></li>
<li class="toctree-l1"><a class="reference internal" href="nibabel_affines.html">The <code class="docutils literal notranslate"><span class="pre">nibabel.affines</span></code> module</a></li>
<li class="toctree-l1"><a class="reference internal" href="nibabel_apply_affine.html">Applying coordinate transforms with <code class="docutils literal notranslate"><span class="pre">nibabel.affines.apply_affine</span></code></a></li>
<li class="toctree-l1"><a class="reference internal" href="map_coordinates.html">General resampling between images with <code class="docutils literal notranslate"><span class="pre">scipy.ndimage.map_coordinates</span></code></a></li>
<li class="toctree-l1"><a class="reference internal" href="image_header_and_affine.html">The image header and affine</a></li>
<li class="toctree-l1"><a class="reference internal" href="images_and_affines.html">Images and affines</a></li>
<li class="toctree-l1"><a class="reference internal" href="rotation_2d_3d.html">Rotations and rotation matrices</a></li>
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<div id="jb-print-docs-body" class="onlyprint">
<h1>Vector and matrix dot products, “np.outer”</h1>
<!-- Table of contents -->
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<h2> Contents </h2>
</div>
<nav aria-label="Page">
<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#vector-dot-products">Vector dot products</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#matrix-dot-products">Matrix dot products</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-outer-product">The outer product</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#dot-vectors-and-matrices">Dot, vectors and matrices</a></li>
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<article class="bd-article" role="main">
<section class="tex2jax_ignore mathjax_ignore" id="vector-and-matrix-dot-products-np-outer">
<h1>Vector and matrix dot products, “np.outer”<a class="headerlink" href="#vector-and-matrix-dot-products-np-outer" title="Permalink to this heading">#</a></h1>
<p>Our standard imports to start:</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="c1"># Display array values to 6 digits of precision</span>
<span class="n">np</span><span class="o">.</span><span class="n">set_printoptions</span><span class="p">(</span><span class="n">precision</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span> <span class="n">suppress</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<section id="vector-dot-products">
<h2>Vector dot products<a class="headerlink" href="#vector-dot-products" title="Permalink to this heading">#</a></h2>
<p>If I have two vectors <span class="math notranslate nohighlight">\(\vec{a}\)</span> with elements <span class="math notranslate nohighlight">\(a_0, a_1, ...
a_{n-1}\)</span>, and <span class="math notranslate nohighlight">\(\vec{b}\)</span> with elements <span class="math notranslate nohighlight">\(b_0, b_1, ... b_{n-1}\)</span>
then the <a class="reference external" href="https://en.wikipedia.org/wiki/Dot_product">dot product</a> is
defined as:</p>
<div class="math notranslate nohighlight">
\[
\vec{a} \cdot \vec{b} = \sum_{i=0}^{n-1} a_ib_i = a_0b_0 + a_1b_1 + \cdots + a_{n-1}b_{n-1}
\]</div>
<p>In code:</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">15</span><span class="p">)</span>
<span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>130
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># The same thing as</span>
<span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="p">)</span> <span class="c1"># Elementwise multiplication</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>130
</pre></div>
</div>
</div>
</div>
<p><code class="docutils literal notranslate"><span class="pre">dot</span></code> is also a <em>method</em> of the NumPy array object, and using the method can
be neater and easier to read:</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">a</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>130
</pre></div>
</div>
</div>
</div>
<p>Better still, the matrix multiplication operator <code class="docutils literal notranslate"><span class="pre">@</span></code> implies a dot product between two vectors:</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">a</span> <span class="o">@</span> <span class="n">b</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>130
</pre></div>
</div>
</div>
</div>
</section>
<section id="matrix-dot-products">
<h2>Matrix dot products<a class="headerlink" href="#matrix-dot-products" title="Permalink to this heading">#</a></h2>
<p>Matrix multiplication operates by taking dot products of the rows of the first
array (matrix) with the columns of the second.</p>
<p>Let’s say I have a matrix <span class="math notranslate nohighlight">\(\mathbf{X}\)</span>, and <span class="math notranslate nohighlight">\(\vec{X_{i,:}}\)</span> is row
<span class="math notranslate nohighlight">\(i\)</span> in <span class="math notranslate nohighlight">\(\mathbf{X}\)</span>. I have a matrix <span class="math notranslate nohighlight">\(\mathbf{Y}\)</span>, and
<span class="math notranslate nohighlight">\(\vec{Y_{:,j}}\)</span> is column <span class="math notranslate nohighlight">\(j\)</span> in <span class="math notranslate nohighlight">\(\mathbf{Y}\)</span>. The output
matrix <span class="math notranslate nohighlight">\(\mathbf{Z} = \mathbf{X} \mathbf{Y}\)</span> has entry <span class="math notranslate nohighlight">\(Z_{i,j} =
\vec{X_{i,:}} \cdot \vec{Y_{:, j}}\)</span>.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span>
<span class="n">X</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[0, 1, 2],
[3, 4, 5]])
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="mi">9</span><span class="p">,</span> <span class="mi">10</span><span class="p">],</span> <span class="p">[</span><span class="mi">11</span><span class="p">,</span> <span class="mi">12</span><span class="p">]])</span>
<span class="n">Y</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[ 7, 8],
[ 9, 10],
[11, 12]])
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">@</span> <span class="n">Y</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[ 31, 34],
[112, 124]])
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span> <span class="o">@</span> <span class="n">Y</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>31
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">@</span> <span class="n">Y</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>112
</pre></div>
</div>
</div>
</div>
</section>
<section id="the-outer-product">
<h2>The outer product<a class="headerlink" href="#the-outer-product" title="Permalink to this heading">#</a></h2>
<p>We can use the rules of matrix multiplication for row vectors and column
vectors.</p>
<p>A row vector is a 2D vector where the first dimension is length 1.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">row_vector</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
<span class="nb">print</span><span class="p">(</span><span class="n">row_vector</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">row_vector</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(1, 3)
</pre></div>
</div>
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[1, 3, 2]])
</pre></div>
</div>
</div>
</div>
<p>A column vector is a 2D vector where the second dimension is length 1.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">col_vector</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">]])</span>
<span class="nb">print</span><span class="p">(</span><span class="n">col_vector</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">col_vector</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(3, 1)
</pre></div>
</div>
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[2],
[0],
[1]])
</pre></div>
</div>
</div>
</div>
<p>We know what will happen if we matrix multiply the row vector and the column
vector:</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">row_vector</span> <span class="o">@</span> <span class="n">col_vector</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[4]])
</pre></div>
</div>
</div>
</div>
<p>What happens when we matrix multiply the column vector by the row vector? We
know this will work because we are multiplying a 3 by 1 array by a 1 by 3
array, so this should generate a 3 by 3 array:</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">col_vector</span> <span class="o">@</span> <span class="n">row_vector</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[2, 6, 4],
[0, 0, 0],
[1, 3, 2]])
</pre></div>
</div>
</div>
</div>
<p>This arises from the rules of matrix multiplication, except there is only one
row * column pair making up each of the output elements:</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">col_vector</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">row_vector</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">col_vector</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">row_vector</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">col_vector</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">row_vector</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[2 6 4]]
[[0 0 0]]
[[1 3 2]]
</pre></div>
</div>
</div>
</div>
<p>This (M by 1) vector matrix multiply with a (1 by N) vector is also called the
<em>outer product</em> of two vectors. We can generate the same thing from 1D
vectors, by using the numpy <code class="docutils literal notranslate"><span class="pre">np.outer</span></code> function:</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">outer</span><span class="p">(</span><span class="n">col_vector</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">row_vector</span><span class="o">.</span><span class="n">ravel</span><span class="p">())</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[2, 6, 4],
[0, 0, 0],
[1, 3, 2]])
</pre></div>
</div>
</div>
</div>
</section>
<section id="dot-vectors-and-matrices">
<span id="dot-vectors-matrices"></span><h2>Dot, vectors and matrices<a class="headerlink" href="#dot-vectors-and-matrices" title="Permalink to this heading">#</a></h2>
<p>Unlike MATLAB, Python has one-dimensional vectors. For example, if I slice a
column out of a 2D array of shape (M, N), I do not get a column vector, shape
(M, 1), I get a 1D vector, shape (M,):</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
<span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span>
<span class="n">v</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([0, 3])
</pre></div>
</div>
</div>
</div>
<p>Because the 1D vector has lost the idea of being a column rather than a row in
a matrix, it is no longer unambiguous what <span class="math notranslate nohighlight">\(v \cdot \mathbf{X}\)</span> means. It
could mean a dot product of a row vector shape (1, M) with a matrix shape
(M, N), which is valid – or a dot product of a row vector (M, 1) with a
matrix shape (M, N), which is not valid.</p>
<p>If you pass a 1D vector into the <code class="docutils literal notranslate"><span class="pre">dot</span></code> function or method, or use the <code class="docutils literal notranslate"><span class="pre">@</span></code>
matrix multiplier, NumPy assumes you mean it to be a row vector on the left,
and a column vector on the right, which is nearly always what you intended:</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># 1D vector is row vector on the left hand side of dot / matrix multiply.</span>
<span class="n">v</span> <span class="o">@</span> <span class="n">X</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([ 9, 12, 15])
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># 1D vector is column vector on the right hand side of dot / matrix multiply.</span>
<span class="n">w</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">X</span> <span class="o">@</span> <span class="n">w</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([2, 2])
</pre></div>
</div>
</div>
</div>
<p>Notice that, in both cases, <code class="docutils literal notranslate"><span class="pre">@</span></code> returns a 1D result.</p>
<p>It sometimes helps to make a 1D vector into a 2D row or column vector, to make
your intention explicit, and preserve the 2D shape of the output:</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Turn 1D vector into explicit row vector</span>
<span class="n">row_v</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="c1"># @ now returns a row vector rather than a 1D vector</span>
<span class="n">row_v</span> <span class="o">@</span> <span class="n">X</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[ 9, 12, 15]])
</pre></div>
</div>
</div>
</div>
</section>
</section>
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