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.