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).