Question Regarding Linear Transformation

[scienceguy288]

I can't figure out how to take the first bite out of this one.

Homework Statement

Let T1: R^2 --> R^2 and T2: R^2 --> R^2 have the indicated properties. Find matrices A, B, and C such that:

T2T1x=Ax, T1T2x=Bx, (T1+T2)x=Cx

Homework Equations

T1e1=(1,3), T1e2=(2,2), T2e1=(-1,1), T2e2=(2,-1)

The Attempt at a Solution

I start by saying that T1e1+T2e1=(T1+T2)e1=(0,4)=Ce1 by adding the two matrices.

Using the same logic, I claim that T2e2+T1e2=(2,2)=(5,1)=Ce2

However, I can't go any further with that because I don't know e1 and e2, don't know T1, T2 (so I can't do the inverse of the transformation). Thus, I cannot find A, B, or C, that is, the sum and products of T1 and T2.

Thanks.

 

[lanedance]

are e1 & e2 the standard basis vectors (1,0) & (0,1)?

 

[Deveno]

it doesn't really matter if the ej are the standard basis or not. they are obviously "some" basis, and we can only give a matrix relative to two bases (the domain basis and the co-domain basis), so we may as well choose {e1,e2} as the domain basis.

 

[scienceguy288]

are e1 & e2 the standard basis vectors (1,0) & (0,1)?

I don't know. Are e1 and e2 standard symbols for the standard basis vectors? Otherwise I think they are just referring to the general vectors {e1, e2}, rather than any specific vector. Perhaps I have to find those vectors first?

it doesn't really matter if the ej are the standard basis or not. they are obviously "some" basis, and we can only give a matrix relative to two bases (the domain basis and the co-domain basis), so we may as well choose {e1,e2} as the domain basis.

That is how I have been approaching the problem thusfar, but as stated in the original problem post, cannot get any further.

 

[scienceguy288]

are e1 & e2 the standard basis vectors (1,0) & (0,1)?

It turns out that this is in fact the case. Still stuck, though...

 

[scienceguy288]

It turns out that this is in fact the case. Still stuck, though...

Nevermind...I got C.

Still running into some trouble finding A and B, but will continue to work on it. If someone can give me a shove in the right direction, that would make my life that much easier...

 

[HallsofIvy]

If you have matrices representing $T_1$ and $T_2$ in the given basis (which are trivial to get), then, as you say, the matrix representing $T_1+ T_2$ is just the sum of those two matrices. And, of course, the matrices representing $T_1T_2$ and $T_2T_1$ are just the products of those two matrices in the given orders.

 

[scienceguy288]

I have solved the problem. Thanks for the help.