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

the bridge from the abstract vector space to the concrete vector space

Exercise time:

Let $V \In \R^3$ be the points that $\{(x_1, x_2, x_3) \mid x_1 + x_2 + x_3=0\}$. Find a basis in $V$, and write the vector $(2,-1,-1)$ in that basis.

Let $W = \R^2$, let $V \to W$ be the map of forgetting coordinate $x_3$. Is this an isomorphism? What's the inverse?

Let $V$ as above, and $W$ be the line generated by vector $(1,2,3)$. Let $f: V \to W$ be the orthogonal projection, sending $v$ to the closest point on $W$. Is this a linear map? How do you show it? What's the kernel? Let $g: W \to V$ be the orthogonal projection. Is it a linear map? What's the relatino between $f$ and $g$?

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

Here is the homework / exercise.

For all the following exercises, if you feel it is too easy, skip it; if you find it interesting and relevant, do it; if you find it too hard, ask about it on discord, let's tackle it together.

1. If you are not familiar with set theory notation and terminology, watch these two short videos notation , how to describe a set (5 min each). If you need to know what is injective / surjective of map of sets, try wikipedia.

• Explain in your own word, what is a set, what is a map. Give examples.
• Can you put 'unrelated things' in a set? Like, a set that consist of two element, $\{\text{your instructor Peng}, \text{a Gorilla}, \text{the Sun}\}$?
• When you define a map $f: A \to B$, where $A,B$ are two sets, can one element in $A$ be sent to two or more elements in $B$?
• try to come up with questions like the above that you don't know about, and ask your classmates or me (or ChatGPT)

2. About linear map. The following functions $f: \R \to \R$, which is a linear map, and which is not?

• $f(x) = |x|$
• $f(x) = (x)^2$
• $f(x) = 3x$
• $f(x) = x+1$.

3. About linear subspace. Let $V = \R^2$, is the following subset $V' \In V$ a subspace? Explain why. You need to check if elements in $V'$ are closed under the vector addition and scalar multiplication. (if you are unfamiliar with the set notation, watch the two videos above.)

• $V' = \{ (x,y) \in \R^2 \mid x+y=0 \}$
• $V' = \{ (x,y) \in \R^2 \mid x+y=1 \}$
• $V' = \{ (x,y) \in \R^2 \mid x>0, y>0 \}$.

4. Matrix manipulations. Write out a $3 \times 2$ matrix $A$, and a $2 \times 2$ matrix $B$, and multiply them together $AB$. Does $BA$ make sense? What do you get?

5. Equivalence relation and equivalence classes. read about it on wiki. Explain it in your own words and give examples. https://en.wikipedia.org/wiki/Equivalence_class

6. Ask someone in your class a question. (the more question the better) Write down the question that you asked and who you asked (could be as simple as “what is $\forall$?” I hope to use this as an excuse to open up talking). You can DM people on discord.

Homework is due on Monday in class. Please write or print out your answer. The homework is graded by completion. It is good opportunity to make mistakes. If you don't do it (not because of it's easy but because you are lazy), then you don't get points.