Part II

So, for the OP and their "Why won’t anyone teach me math?", if you wish, can learn linear algebra like I did and outlined above. Then can effortlessly blow away the other students and even intimidate the prof even in an advanced flunk out course in linear algebra.

Sure, can learn from Nering and Halmos if you want. Also looks good to me is

Hoffman and Kunze, Linear Algebra, Second Edition, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.

apparently now available at

     https://www.math.pku.edu.cn/teachers/anjp/textbook.pdf

There is also a text from G. Strang that is recommended for a first text.

I will try to help get the frustrated Princeton student started:

First, it is fair to say that a good start on linear algebra is just solving a system of linear equations. E.g., given numbers a and b, find the set of all numbers x so that

ax = b

Exercise: Show that depending on the values of a and b, the set of all solutions consists of none, one, or infinitely many values.

The

ax = b

is one linear equation in one unknown, x.

Well for positive integers m and n, we can have m linear equations in n unknowns. So, here are 2 linear equations in 3 unknowns, x, y, and z:

     2x -y + 5z = 7

     x + y - 2z = 3

Why are they called linear? Good, crucial question -- profound issue; will come to that!

For reasons analogous to what saw with

ax = b

again the set of solutions of m linear equations in n unknowns has none, one, or infinitely many solutions.

Exercise: We have already treated the case of m = n = 1. Treat the case of m = 1 when n > 1.

Assume the number of equations m > 1: For finding the set of all solutions, the standard approach is Gauss elimination.

Here is the key idea: If take a number a and multiply one of the m equations by the number a and add it to one of the other of the m equations, then the set of solutions does not change. That is called an elementary row operation (ERO).

Exercise: Argue this point -- it's easy.

So, with Gauss elimination just apply EROs to yield the equations with lots of coefficients 0 that permit reading off the set of all solutions easily.

How to do this? Do an ERO to have a 1 as the coefficient in row 1, column 1 and 0s in the rest of column 1. Now do an ERO to put a 1 in the 2, 2 position and 0s below that. Continue in this way and end up with a triangle of 0s. Now can just read off the set of all solutions.

Here start to see that get a lot more for your time and effort than you expected. Back to

     2x -y + 5z = 7

     x + y - 2z = 3

We rip out the variables x, y, z and write all this as

       /         \   / \     / \
      |  2  -1  5 | | x |   | 7 |
      |           | |   | = |   |
      |  1  +1 -2 | | y |   | 3 |
       \          / |   |    \ /
                    | z |
                     \ /

So we have three matrices, the one on the left has 2 rows and 3 columns so is said to be 2 x 3. The next one has the x, y, and z and is 3 x 1. The one on the right is 2 x 1.

For the first two matrices, we define the matrix product to yield essentially the same thing we had with the 2 equations in 3 unknowns.

So we have just rewritten the 2 equations in 3 unknowns. It turns out, however, that working with the matrices is a huge improvement, step up.

Vector spaces? A matrix with one row and/or one column is called a vector. Sometimes we can be more specific and call the vectors with 1 row dual vectors. Then if we generalize a little we can prove the Riesz representation theorem that gets used in quantum mechanics. Point: Linear algebra is an introduction, simple elementary special case, of a lot more in pure and applied math.

Quite generally we say that a function F is linear if

     F(ax + by) = aF(x) + bF(y)

Here I have deliberately not given a careful definition of the symbols F, a, b, x, and y because (i) linearity is one of the largest pillars of math and (ii) there are lots of cases with different definitions for the F, a, b, x, and y. E.g., in calculus, both differentiation and integration are linear. For more, in electronic engineering and signal processing, every time invariant linear system just modifies the amplitude and phase of sine waves.

The real world and applications of math to it are just awash in linearity.

Well, matrix multiplication is linear! And that's why the equations

     2x -y + 5z = 7

     x + y - 2z = 3

are linear.

Okay, suppose m = n. Then a matrix m x n is square. Given a matrix U, suppose for any x the length of Ux is the same as that of x. Then all U can do is do rigid rotations and reflections. Or suppose given a matrix H where all it does convert a sphere into an ellipsoid with mutually perpendicular axes. Now given a square matrix A, there exist U and H so that

A = HU.

The U is called unitary and the H, Hermitian. In quantum mechanics, the evolution of the wave functions is unitary, and the measurements are Hermitian. Principle components in statistics is based on Hermitian.

The A = HU is the polar decomposition. So, linearity is simple: All the linear A can do is rotate and/or reflect and then stretch and/or contract on mutually perpendicular axes.

Fill in all the details, and that should be 80+% of a first course in linear algebra.

For more, take Gauss elimination and specialize it slightly and get the simplex algorithm of linear programming optimization. At one time, Princeton, along with Berkeley, played a leading role in linear programming. Then linear programming became the core of at least one Nobel prize in economics.

Can use some of the basic theory of linear programming to show the saddle-point theorem of game theory, e.g., was done by different methods by von Neumann.

To defend yourself from the propensity of pure math profs to dump you into the contempt bucket: Move to some advanced, specialized material, find a loose end, state and prove a theorem to tie off the loose end, and publish the result. Can work better than Kryptonite.