Linear transformation 2 x 2 matrix problem

[SYoungblood]

Homework Statement

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

Homework Equations

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See the above notes

The Attempt at a Solution

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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.

Thanks much for your help

SY

 

[Orodruin]

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.

 

[SYoungblood]

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.

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.

 

[SYoungblood]

I think I got it

0 1

1 3

Am I in the ballpark?

SY

 

[Orodruin]

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.

 

[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.

 

[Orodruin]

Yes, that is correct.

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

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

 

[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