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Table of Contents
Review of Linear Algebra
Let's start from scratch again. What is linear algebra?
This is a textbook on linear algebra by Prof Givental.
What is a vector space
Answer 1
row vectors, column vectors, matrices. Let's also review the index notation $a_i = \sum_{j} M_{ij} b_j$
Very concrete, very computable.
Answer 2
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.
Answer 3
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?
Linear Map
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
Basis vectors:
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 $g$?
