Linear transformation and Change of Basis

[Jimmy84]

Homework Statement

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)

Homework Equations

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

 

[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.

 

[Jimmy84]

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.

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

 

[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 $e_1 = (1,0)$.