Review of Linear Algebra

Let's start from scratch again. What is linear algebra?

This is a textbook on linear algebra by Prof Givental.

row vectors, column vectors, matrices. Let's also review the index notation $a_i = \sum_{j} M_{ij} b_j$

Very concrete, very computable.

Geometrical, as we went over in class.

A 2-dimentional vector is something you can draw.

A 3-dim vector, hmm, harder.

how about 4-dim vector? $\infty$-dim one? It doesn't matter the dimension, the rule we obtain from 2 and 3 dimensional one is good enough.

the goofy math prof: a vector space is a set $V$ together with two operations

• scalar multiplication: given a number $c$ and a vector $v \in V$, we need to specify the output $c v \in V$.
• vector addition: given two vectors $v_1, v_2 \in V$, we need to specify the output $v_1 + v_2 \in V$

such that, some obvious conditions should be satisfied.

Why we care about this? Because it is somehow useful.

For example,

• the subspace of $\R^3$ that is perpendicular to $(1,2,3)$ forms a vector space.
• other examples? non-examples?

How does vector space talk to each other? Linear map.

Do an example of stretching, skewing.

Do a non-example of bending a line in $\R^2$.

Kernel, image and cokernel

We didn't quite cover the idea of a quotient space. I will do that next time.