Linear transformations formula

[Aerosion]

Linear transformations...

Homework Statement

Can't figure these things out for my life. Seriously. Here's an example.

consider the basis S={v1, v2, v3} for R^3 where v1=(1,2,1), v2=(2,9,0) and v3=(3,3,4) and let T:R^3-->R^2 be the linear transformation such that:

T(v1)=(1,0) T(v2)=(-1,1) T(v3)=(0,1)

Find formula for T(x1,x2,x3) and use it to find T(7,13,7)

Homework Equations

The Attempt at a Solution

So here I am starting it, writing c1(1,2,1)+c2(2,9,0)-C3(3,3,0) to make C1+2C2+3C3, 2C1+9C2+3C3, C1+C3

c1+2c2+3c3=x1
2c1+4c2+3c3=x2
c1+ +4c3=x3

Now, I'm having trouble solving for c1, c2, and c3. The answers seem to be c1=36x1+8x2+21x3, c2=5x1-x2-3x3, c3=9x1-2x2-5x3. I don't know where those numbers came from. (I was guessing putting it into a matrix and use row reduction, but...no). Hmm? Help?

 

[Mindscrape]

So the question is asking for the matrix that sends (1,2,1), ..., to their respective vectors in lR^2.

A v1 = ?
A v2 = ?
A v3 = ?

It seems like you are trying to find linear combinations of the vectors that will give the transformations. I don't think that is going to work.

If you can find the matrix that performs the transformations, you can use matrix multiplication to find the vectors corresponding to certain coordinates (c1,c2).