# Linear Algebra II

## October 20, 2014

### Lecture 16

Filed under: 2014 Fall — Y.K. Lau @ 5:15 PM

Today we illustrated the use of the First Isomorphism Theorem, please see 1stFT(updated).pdf for the details.

Next we returned to Chapter 9. The motivation is based on

• Every matrix ${A\in M_{n,m}}$ induces a linear transformation ${T_A:{\mathbb R}^n\rightarrow {\mathbb R}^m}$ (defined by ${T_A(x)=Ax}$),
• Every linear transformation ${f:{\mathbb R}^n\rightarrow {\mathbb R}^m}$ can be represented by a matrix ${A\in M_{n,m}}$, in the sense that ${f=T_A}$. (See Lect16a.pdf)

Now we hope to extend the result to general vector spaces (i.e. to represent a general linear transformation by a matrix).

The goal is clear and natural but how to do it is another matter.

First, how do we link a general vector to a column vector? This results in the concept of the coordinate vector with respect to a basis. We did this today. See Lect16b.pdf

Second what do we mean by representing a general linear transformation by a matrix? We need a proper setting to describe what we mean. This will be done in the next lecture.

Some classmates suggested more exercises in the survey last week. Indeed, the workload from the tutorial, assignments and textbook exercise questions are pretty high, so I choose to work out some questions in the textbook for your study. See L16-ace.pdf).

Remark. All the pdf’s are stored in the folder “Slides” in moodle.