<story><title>Why won’t anyone teach me math?</title><url>https://www.dailyprincetonian.com/article/2022/02/stem-intro-courses-humanites-exploration</url></story><parent_chain><item><author>QuercusMax</author><text>I took linear algebra the same semester as computer graphics - the first half of CG was all about 2D stuff - Bresenham&amp;#x27;s line-drawing algorithm, things like that. First half of LA was pretty much everything I needed to know to understand matrix operations in OpenGL, which was the second half of CG. Worked great! I still remember most of the relevant stuff 20 years later.&lt;p&gt;The second half of linear algebra was a bunch of stuff about eigenvectors and eigenvalues. We never were given an explanation why you&amp;#x27;d care about them, and I still have no idea why you&amp;#x27;d care about them.</text></item><item><author>Animats</author><text>Here&amp;#x27;s the class description for MAT 202, &amp;quot;Linear Algebra with Applications&amp;quot;  at Princeton.[1] Here&amp;#x27;s a problem set with answers.[2]&lt;p&gt;It&amp;#x27;s painful to read. Problem 2: &amp;quot;Find the matrix A describing the following linear transformation from R2 to R&amp;#x27;2 : First rotate clockwise by π&amp;#x2F;6, then scale by a factor of 2, then reflect about the x2 axis.&amp;quot; Anyone who programs video games recognizes that as a stack of transformations. You set up the matrix for the rotate, the scale, and the reflection (which is a scale of -1 on one axis). You multiply the matrices to get the result. This is a pain to do with pencil and paper. GPUs have hardware for it.&lt;p&gt;Books on graphics programming cover this problem. But they usually do it much better, with graphical examples.
The presentation here is done very abstractly, without any motivation.
Most of linear algebra has graphical representations. It takes a useful and relatively easy area of math and makes it harder.&lt;p&gt;This is apparently on purpose.  The class description says:&lt;p&gt;&lt;i&gt;The calculations are relatively simple once you understand what you need to compute, but it can take time to master the abstract concepts well enough to understand what that might be. The challenge in 201 (a calculus class) is usually how to finish a problem as the technical complications mount, whereas in 202 the challenge is often in seeing how to start the problem.&lt;/i&gt;&lt;p&gt;Right. In calculus, problems usually involve trying to integrate something, where you get stuck and need to transform the problem in some non-obvious way to make forward progress. It&amp;#x27;s puzzle-solving. In linear algebra, the actual operations are mostly matrix adds and multiplies. It&amp;#x27;s setting up the problem that&amp;#x27;s hard.&lt;p&gt;It&amp;#x27;s annoying to see this for a useful area of applied math. If you&amp;#x27;re headed for abstract algebra, where intuition breaks down, this approach might be useful. But if you want to use ordinary linear systems to get work done, which is common in engineering, it&amp;#x27;s not.&lt;p&gt;[1] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;www.math.princeton.edu&amp;#x2F;undergraduate&amp;#x2F;placement&amp;#x2F;MAT202&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;www.math.princeton.edu&amp;#x2F;undergraduate&amp;#x2F;placement&amp;#x2F;MAT20...&lt;/a&gt;&lt;p&gt;[2] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;www.math.princeton.edu&amp;#x2F;sites&amp;#x2F;default&amp;#x2F;files&amp;#x2F;2018-04&amp;#x2F;MAT202Sample%20Problems.pdf&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;www.math.princeton.edu&amp;#x2F;sites&amp;#x2F;default&amp;#x2F;files&amp;#x2F;2018-04&amp;#x2F;M...&lt;/a&gt;</text></item></parent_chain><comment><author>klodolph</author><text>Eigenvectors and eigenvalues are (among other things) connected to something called diagonalization. In math, this is incredibly useful because it lets you describe a matrix as being put together from simpler, easy-to-understand matrixes.&lt;p&gt;Eigenvectors and eigenvalues are also a major focus of quantum mechanics. For example, when you measure the energy level of an electron in a quantum mechanical system, the measurement itself is a linear operator on the underlying wavefunction (linear operator = think &amp;quot;like a matrix&amp;quot;). The eigenvalues are the different energy levels of the electron, and the eigenvectors are the wavefunctions which are states of the electron with the corresponding energy level. You can experimentally verify that the eigenvalues correspond to spectral lines (the rainbow you see when you look at the substance through diffraction grating).&lt;p&gt;There are a ton of other things going on with eigenvalues and eigenvectors, they&amp;#x27;re used all over the place. If you want to understand a Markov process, for example, it can be described in terms of linear equations, and if it has a steady-state, it&amp;#x27;s an eigenvector.</text></comment>
<story><title>Why won’t anyone teach me math?</title><url>https://www.dailyprincetonian.com/article/2022/02/stem-intro-courses-humanites-exploration</url></story><parent_chain><item><author>QuercusMax</author><text>I took linear algebra the same semester as computer graphics - the first half of CG was all about 2D stuff - Bresenham&amp;#x27;s line-drawing algorithm, things like that. First half of LA was pretty much everything I needed to know to understand matrix operations in OpenGL, which was the second half of CG. Worked great! I still remember most of the relevant stuff 20 years later.&lt;p&gt;The second half of linear algebra was a bunch of stuff about eigenvectors and eigenvalues. We never were given an explanation why you&amp;#x27;d care about them, and I still have no idea why you&amp;#x27;d care about them.</text></item><item><author>Animats</author><text>Here&amp;#x27;s the class description for MAT 202, &amp;quot;Linear Algebra with Applications&amp;quot;  at Princeton.[1] Here&amp;#x27;s a problem set with answers.[2]&lt;p&gt;It&amp;#x27;s painful to read. Problem 2: &amp;quot;Find the matrix A describing the following linear transformation from R2 to R&amp;#x27;2 : First rotate clockwise by π&amp;#x2F;6, then scale by a factor of 2, then reflect about the x2 axis.&amp;quot; Anyone who programs video games recognizes that as a stack of transformations. You set up the matrix for the rotate, the scale, and the reflection (which is a scale of -1 on one axis). You multiply the matrices to get the result. This is a pain to do with pencil and paper. GPUs have hardware for it.&lt;p&gt;Books on graphics programming cover this problem. But they usually do it much better, with graphical examples.
The presentation here is done very abstractly, without any motivation.
Most of linear algebra has graphical representations. It takes a useful and relatively easy area of math and makes it harder.&lt;p&gt;This is apparently on purpose.  The class description says:&lt;p&gt;&lt;i&gt;The calculations are relatively simple once you understand what you need to compute, but it can take time to master the abstract concepts well enough to understand what that might be. The challenge in 201 (a calculus class) is usually how to finish a problem as the technical complications mount, whereas in 202 the challenge is often in seeing how to start the problem.&lt;/i&gt;&lt;p&gt;Right. In calculus, problems usually involve trying to integrate something, where you get stuck and need to transform the problem in some non-obvious way to make forward progress. It&amp;#x27;s puzzle-solving. In linear algebra, the actual operations are mostly matrix adds and multiplies. It&amp;#x27;s setting up the problem that&amp;#x27;s hard.&lt;p&gt;It&amp;#x27;s annoying to see this for a useful area of applied math. If you&amp;#x27;re headed for abstract algebra, where intuition breaks down, this approach might be useful. But if you want to use ordinary linear systems to get work done, which is common in engineering, it&amp;#x27;s not.&lt;p&gt;[1] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;www.math.princeton.edu&amp;#x2F;undergraduate&amp;#x2F;placement&amp;#x2F;MAT202&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;www.math.princeton.edu&amp;#x2F;undergraduate&amp;#x2F;placement&amp;#x2F;MAT20...&lt;/a&gt;&lt;p&gt;[2] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;www.math.princeton.edu&amp;#x2F;sites&amp;#x2F;default&amp;#x2F;files&amp;#x2F;2018-04&amp;#x2F;MAT202Sample%20Problems.pdf&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;www.math.princeton.edu&amp;#x2F;sites&amp;#x2F;default&amp;#x2F;files&amp;#x2F;2018-04&amp;#x2F;M...&lt;/a&gt;</text></item></parent_chain><comment><author>l33t2328</author><text>Eigenvectors tell you which vectors are taken to a scalar multiple of themselves via a linear map. Eigenvalues tell you what those scalars are. That means some subspace is fixed by your linear map.</text></comment>