Homework Help: Linear transformation 2 x 2 matrix problem

1. Aug 1, 2015

SYoungblood

1. The problem statement, all variables and given/known data

Find a 2 x 2 matrix that maps e1 to –e2 and e2 to e1+3e2

2. Relevant equations

See the above notes

3. The attempt at a solution

I am making a pig's ear out of this one.

I think I can get e1 to –e2

3 -1

1 -3

but as far as getting it to reconcile a matrix like the above to match both conditions is kicking my posterior.

SY

2. Aug 2, 2015

Orodruin

Staff Emeritus
The matrix you quote does not map e1 to -e2. I recommend that you start by writing down the most general matrix and then see how it acts on e1 and e2 respectively. From there you can identify the elements of the matrix.

3. Aug 2, 2015

SYoungblood

Here's a simple 2x2 mattrix --
0 1

-1 0
So, if I multiply that by [1 over 0], I get 1 * [0 over -1} + 0 [1 over 0] = [0 over -1].

Multiplying by our T vector [1 over 0], e1has become -e2 by taking the former e2 and basically multiplying this by -1 and turning it upside down.

I am having problems wrapping my mind around this one and I am trying to break it down Barney style.

4. Aug 2, 2015

SYoungblood

I think I got it

0 1

1 3

Am I in the ballpark?

SY

5. Aug 2, 2015

Orodruin

Staff Emeritus
In the ballpark, yes. That matrix maps e1 to e2 and e2 to e1+3e2, ie, you are missing a minus sign in the mapping of e1.

6. Aug 2, 2015

SYoungblood

Aha.

0 1

-1 3

To use your tag line, a child of five might understand this, but I am not a child of five.

7. Aug 2, 2015

Orodruin

Staff Emeritus
Yes, that is correct.

I would lend you my niece, but she turned six the day before yesterday. ;)

8. Aug 2, 2015

SYoungblood

Geez, now that I have the answer, it is amazing how easy this question was.

Thank you very much for your help, I greatly appreciate it.

SY