The Philosophy of Linear Algebra (2021)
These articles remind me that one should look in the right places for the right respective knowledge.<p>That article is not at all an exposition of what the Philosophy of Linear Algebra is.<p>&gt;While nowadays most people learn linear algebra through the perspective of vectors and transformations, the matrix and determinant are much older than the ideas of vectors in a multi-dimensional space.<p>That is not true. Anyway, what is the article about? According to the title it&#x27;s about linear algebra. Then, the basic objects are vectors, and the interesting topic of discussion are maps. Some maps can be represented by matrices. That&#x27;s it.<p>Yet, the article begins by describing matrices. Is it like those fiction books who randomize the order of the chapters to make them more interesting?<p>Or is the article supposed to be about matrices in general, not about linear algebra? (Niche) uses of matrices outside of linear algebra do exist.<p>&gt;An upper triangular matrix conveys a whole lot more than just the fact that the numbers form a triangle; it tells you that the matrix is describing a nonsingular system that is solvable.<p>That is not true in general.<p>If the matrix is an upper triangular matrix AND all the coefficients in the diagonal are non-zero, THEN it&#x27;s regular (so invertible, so non-singular).<p>But philosophically, WHY does it do that?<p>The answer is because you are basically trying to invert the matrix but you notice half-way through that you can already use the upper triangular matrix to get the result with non-matrix means, so you just stop there instead (it&#x27;s a lot less work). It can be shown that the kind of manipulations you do to get from the triangular matrix to the inverse matrix are all conserving the determinant, so you&#x27;re in the clear.<p>That is slightly interesting from a philosophical standpoint.<p>&gt;The diagonal matrix describes a system with elements that only operate on themselves,<p>It just means the elements don&#x27;t mix the different axes of the multiplicand in the result. Coordinate along axis one in the result = coordinate along axis one multiplied by diagonal element one. And so on, for two, three.<p>&gt;symmetric matrix tells the story of a system that is bi-directional<p>Argh. It just means that the eigenvectors are all orthogonal to each other and the eigenvalues are all real. It&#x27;s nice to have these properties (for example because it&#x27;s intuitive to have orthogonal coordinate axes), so it&#x27;s often used for calculations.<p>Also, eigenvectors and eigenvalues are interesting because they give you a better basis for your calculation (literally).<p>If you choose your coordinate axes wrong, it will tell you the &quot;better&quot; ones (the one intrinsic to the problem--so it&#x27;s much easier to calculate things there).<p>To summarize, to save others the work: that article is not about the philosophy of linear algebra. It&#x27;s just showing a small example with matrices and not explaining any WHYs in the process. Where it does explain things, it&#x27;s wrong multiple times.<p>----<p>Maybe a small real Philosophy of Linear Algebra here:<p>The real Philosophy of Linear Algebra is that there are a lot of systems that are linear, therefore it&#x27;s nice to have a complete mathematical theory for systems of linear equations. That&#x27;s it. It&#x27;s not magical, it&#x27;s not general enough (<i>cough</i> non-linear systems), and (looked at it right) it&#x27;s simple.<p>Linear algebra&#x27;s objects of discussion are vectors and maps.<p>What vectors are everyone learned in school. Suffice to say vectors are any objects which can be added together, multiplied by a scalar, and then form a commutative ring with respect to addition; and multiplication and addition distribute over in the usual way (like for real numbers). Notably, there&#x27;s no division! This is not a complete definition, but you get the idea.<p>(It is also useful, but not necessary, to introduce a so-called &quot;inner&quot; product. Inner because the result stays inside the vector space. Nevermind!)<p>In any case, you can then have equations with vectors in them. Like 5 v⃗ = w⃗, where v⃗, w⃗ are vectors.<p>This works like any other system of equations: there are not enough equations for those two unknowns v⃗ and w⃗.<p>So what about this system of equations: { 5 v⃗ = w⃗, 5 v⃗ = w⃗ } ?  That&#x27;s silly you say?<p>Ok then, what about { 5 v⃗ = w⃗, 10 v⃗ = 2 w⃗ } ? Still doesn&#x27;t work?<p>So what&#x27;s the rule here? Turns out it has to do with those upper triangular matrices.<p>A system that has no interesting solution (that is not 0⃗):<p><pre><code>   5 v⃗ = w⃗
   5 v⃗ = 2 w⃗

   w⃗ = 2 w⃗
   0⃗ = 2 w⃗ - w⃗
   0⃗ = w⃗
</code></pre>
Booring.<p>Anyway, let&#x27;s introduce functions on vectors, so-called &quot;maps&quot; (here, f and g are maps).<p><pre><code>   f(v⃗) := 5 v⃗
   g(v⃗) := (5&#x2F;2) v⃗

   f(v⃗) = w⃗
   g(v⃗) = w⃗
</code></pre>
THAT is the basic philosophy of linear algebra.<p>Now for some more practical stuff:<p>It&#x27;s annoying to have those abstract v⃗ and w⃗ be around. In a lot of cases, you are only interested in coordinates with respect to some basis (the basis is a set of axes that you choose--usually your axes of measurement. That is the axes your measurement devices measure along). It can be shown that the number of basis vectors you need for that (to represent any abstract vector!) is equal to the dimension of the vector space, usually 3.<p>Let the basis be {b⃗_1, b⃗_2, b⃗_3}. These basis vectors obviously must be independent, so b⃗_1 := b⃗_2 is out because that&#x27;s cheating. b⃗_1 := 2 b⃗_2 ? Out. b⃗_1 := 2 b⃗_2 + 3 b⃗_3 ? Stop it already.<p>Now you can find numbers q, r, s such that v⃗ = q b⃗_1 + r b⃗_2 + s b⃗_3.<p>If you want to be more conserative with your letter consumption, better is to call q and r and s: v^1, v^2, v^3 instead (it&#x27;s just a notation, and the ^1 and ^2 and ^3 are part of the name for now).<p>So<p><pre><code>   v⃗ = v^1 b⃗_1 + v^2 b⃗_2 + v^3 b⃗_3.
   w⃗ = w^1 b⃗_1 + w^2 b⃗_2 + w^3 b⃗_3.
</code></pre>
Or in non-annoying notation<p><pre><code>   v⃗ = ∑_i v^i b⃗_i
   w⃗ = ∑_i w^i b⃗_i
</code></pre>
Or in even less annoying notation, let the sum be implied when an index repeats in a term<p><pre><code>   v⃗ = v^i b⃗_i
   w⃗ = w^i b⃗_i
</code></pre>
Ahhh. Much better.<p>If the f is a LINEAR map, then you can do this (by definition, which you have to look up):<p><pre><code>   f(∑_i v^i b⃗_i) = ∑_i v^i f(b⃗_i)
</code></pre>
So in order to specify the linear map f completely, you only need {_i f(b⃗_i) }, i.e. how f acts on the individual basis vectors each. That&#x27;s it.<p>So for f and g in non-annoying notation:<p><pre><code>  f(v^i b⃗_i) = v^i f(b⃗_i)
  g(v^i b⃗_i) = v^i g(b⃗_i)
</code></pre>
Also,<p><pre><code>  f(v⃗) = w⃗ = w^i b⃗_i
  g(v⃗) = w⃗ = w^i b⃗_i
</code></pre>
So<p><pre><code>  f(v^i b⃗_i) = v^i f(b⃗_i) = w^i b⃗_i
  g(v^i b⃗_i) = v^i g(b⃗_i) = w^i b⃗_i
</code></pre>
So<p><pre><code>  v^i f(b⃗_i) = w^i b⃗_i
  v^i g(b⃗_i) = w^i b⃗_i
</code></pre>
And NOW we can go to matrices.<p>Those b⃗_i are annoying, and the f(b⃗_i) are even more annoying.<p>So let&#x27;s get rid of them.<p>Let c⃗_i := f(b⃗_i).
Let d⃗_i := g(b⃗_i).<p>Now we&#x27;d like all those abstract basis vectors to disappear. But c⃗_i and b⃗_i are different bases (if they are bases at all--neeevermind for now :P), we&#x27;d like these to be specified only with the b⃗_i. Let c⃗_i = c^j_i b⃗_j<p><pre><code>   v^i c⃗_i = w^i b⃗_i
   v^i c^j_i b⃗_j = w^i b⃗_i
   v^i c^j_i b⃗_j = w^i δ^j_i b⃗_j

   c^j_i v^i = w^j

   c := (_i (_j c^j_i)) That&#x27;s the first matrix!
   v := (_i v^i)        That&#x27;s a column &quot;vector&quot;--a matrix
   w := (_i w^i)        That&#x27;s a column &quot;vector&quot;--a matrix
</code></pre>
Note: δ^j_i := {1 if i=j, otherwise 0}<p><pre><code>   c v = w
</code></pre>
Likewise for the g, with the d⃗_i killed and replaced by the b⃗_i.<p>And now we are at the philosophically completely uninteresting result. Good job.