# Linear transformation and Change of Basis

1. Jul 23, 2012

### Jimmy84

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

Greetings, I have been stuck with this problem for a while, I thought maybe someone could give me some advice about it. Thanks a lot in advance.

If T is a linear transformation that goes from R^2 to R^2 given that T(v1)= -2v2 -v1 and
T(v2)=3v2.

and B = v1=(1,1) , v2=(1,-1)

Find T with respect to the base B and T with respect to Nat, (the Natural Base)

2. Relevant equations

3. The attempt at a solution

I found T with respect to B by inspection

-1 0
-2 3

How can I find T with respect to the natural base?

Thanks

2. Jul 23, 2012

### Robert1986

Write the two natural basis vectors in terms of $v_1,v_2$ and then see what this transformation does to them. Once you know that, then you know how to find the matrix.

3. Jul 24, 2012

### Jimmy84

Do you mean to write 1,0 and 0,1 as a linear combination of v1 and v2 ? how can I see what the transformation does to them when I'm not given the transformation?

Thanks

4. Jul 24, 2012

### Robert1986

Well, let's say $e_1 = c_1v_1 + c_2v_2$. Then, $T(e_1) = T(c_1v_1 + c_2v_2)$. Now, use the linearity of $T$ and what you know about $T(v_1)$ and $T(v_2)$ to calculate $T(e_1)$.

EDIT:
I don't know what terms your book uses, but I mean that $e_1 = (1,0)$.

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