1. ## Linear Map Problem

I Just started the course. I need more example problem.

I know how to find a formula for f, but the question 1 blew, given what is satisfied, how do i compute the f ? What is the meaning of that?

what is a linea transformation? my instructor just threw problems, hardly explains..

http://www.Photo-Host.org/img/589464math207.png

2. Originally Posted by yzc717
I Just started the course. I need more example problem.

I know how to find a formula for f, but the question 1 blew, given what is satisfied, how do i compute the f ? What is the meaning of that?

what is a linea transformation? my instructor just threw problems, hardly explains..
I think you need to provide more information (I can't see your image file as photo hosting sites are blocked on this computer), but:

A linear transformation $\displaystyle f$ on a vector space $\displaystyle S$ over a field $\displaystyle F$ (usualy $\displaystyle \mathbb{R}$ or $\displaystyle \mathbb{C}$) to another vector space over the same field, is a transformation such that:

$\displaystyle \forall u,v \in S, \alpha, \beta \in F;\ \ f(\alpha u + \beta v) =\alpha f(u)+\beta f(v)$

RonL

3. can u see the photo now?

how do i prove the problem 4?

4. Originally Posted by yzc717
can u see the photo now?

how do i prove the problem 4?
Yes, I can now see the picture.

The $\displaystyle i$ -th diagonal element of $\displaystyle AB$ is:

$\displaystyle d_i=\sum_j A_{i,j}B_{j,i}$

Hence the trace of $\displaystyle AB$ is:

$\displaystyle tr(AB)=\sum_i d_i=\sum_i \sum_j A_{i,j}B_{j,i}$

Now these are finite sums so the order of summation can be reversed, so:

$\displaystyle tr(AB)=\sum_j \sum_i A_{i,j}B_{j,i}=\sum_j \sum_i B_{j,i}A_{i,j}=tr(BA)$

RonL