<story><title>An Intuitive Guide to Linear Algebra (2012)</title><url>https://betterexplained.com/articles/linear-algebra-guide/</url></story><parent_chain><item><author>xscott</author><text>I think this is spot on.  Depending on what you&amp;#x27;re doing, a matrix can be:&lt;p&gt;&lt;pre&gt;&lt;code&gt;    - A linear transformation
    - A basis set of column vectors
    - A set of equations (rows) to be solved
       - (your example: parity equations for coding theory)
    - The covariance of elements in a vector space
    - The Hessian of a function for numerical optimization
    - The adjacency representation of a graph
    - Just a 2D image (compression algorithms)
    ... (I&amp;#x27;m sure there are plenty of others)
&lt;/code&gt;&lt;/pre&gt;
For some of these, the matrix is really just a high dimensional number.  You (rarely?) never think of covariance in a Kalman filter as a linear transform, but you still need to take its Eigen vectors if you want to draw ellipses.</text></item><item><author>nightcracker</author><text>&amp;gt; Linear algebra is really about linear transformations of vector spaces, which is not captured in this blog post.&lt;p&gt;I... disagree. Some of linear algebra is about that. And it&amp;#x27;s probably a good way to view it that way when learning.&lt;p&gt;But some of my current work (coding theory) involves linear
algebra over finite fields. We use results from linear algebra, and interpret our problem using matrices, but really
at no point are we viewing what we&amp;#x27;re doing as transforming a vector space, we&amp;#x27;re just solving equations with unknowns.</text></item><item><author>vecter</author><text>This is ok but nothing is as intuitive as 3B1B&amp;#x27;s series on YouTube that has been posted hundreds of times on HN [0].&lt;p&gt;Linear algebra is really about linear transformations of vector spaces, which is not captured in this blog post.&lt;p&gt;[0] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;www.youtube.com&amp;#x2F;watch?v=fNk_zzaMoSs&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;www.youtube.com&amp;#x2F;watch?v=fNk_zzaMoSs&lt;/a&gt;</text></item></parent_chain><comment><author>obastani</author><text>The first three can reasonably be thought of as defining linear transformations. For linear systems of equations A x = b, x is an unknown vector in the input space that is mapped by A to b.&lt;p&gt;Both covariance matrices and Hessians are more naturally thought of as tensors, not matrices (and therefore not linear transformations). That is, they take in two vectors as input and produce a single real number as output.&lt;p&gt;As for graph adjacency matrix, this can actually be thought of as a linear transformation on the vector space where the basis vectors correspond to nodes in the graph. Linear combinations of these basis vectors correspond to probability distributions over the graph (if properly normalized).&lt;p&gt;2D images... Yes, these cannot really be interpreted as linear transformations. But I&amp;#x27;d say these aren&amp;#x27;t really matrices in the mathematical sense.</text></comment>
<story><title>An Intuitive Guide to Linear Algebra (2012)</title><url>https://betterexplained.com/articles/linear-algebra-guide/</url></story><parent_chain><item><author>xscott</author><text>I think this is spot on.  Depending on what you&amp;#x27;re doing, a matrix can be:&lt;p&gt;&lt;pre&gt;&lt;code&gt;    - A linear transformation
    - A basis set of column vectors
    - A set of equations (rows) to be solved
       - (your example: parity equations for coding theory)
    - The covariance of elements in a vector space
    - The Hessian of a function for numerical optimization
    - The adjacency representation of a graph
    - Just a 2D image (compression algorithms)
    ... (I&amp;#x27;m sure there are plenty of others)
&lt;/code&gt;&lt;/pre&gt;
For some of these, the matrix is really just a high dimensional number.  You (rarely?) never think of covariance in a Kalman filter as a linear transform, but you still need to take its Eigen vectors if you want to draw ellipses.</text></item><item><author>nightcracker</author><text>&amp;gt; Linear algebra is really about linear transformations of vector spaces, which is not captured in this blog post.&lt;p&gt;I... disagree. Some of linear algebra is about that. And it&amp;#x27;s probably a good way to view it that way when learning.&lt;p&gt;But some of my current work (coding theory) involves linear
algebra over finite fields. We use results from linear algebra, and interpret our problem using matrices, but really
at no point are we viewing what we&amp;#x27;re doing as transforming a vector space, we&amp;#x27;re just solving equations with unknowns.</text></item><item><author>vecter</author><text>This is ok but nothing is as intuitive as 3B1B&amp;#x27;s series on YouTube that has been posted hundreds of times on HN [0].&lt;p&gt;Linear algebra is really about linear transformations of vector spaces, which is not captured in this blog post.&lt;p&gt;[0] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;www.youtube.com&amp;#x2F;watch?v=fNk_zzaMoSs&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;www.youtube.com&amp;#x2F;watch?v=fNk_zzaMoSs&lt;/a&gt;</text></item></parent_chain><comment><author>smallnamespace</author><text>A covariance matrix naturally transforms from the measured space to a space where things are approximately unit Gaussian distributed. This is identical to the Z transform in 1D case.&lt;p&gt;This can be useful in, say, exotic options trading - a natural unit of measurement is how many ‘vols’ an underlier has moved, e.g. a 10-vol move is very large.</text></comment>