Linear Transformations of Matrices

[schmidtc89]

Homework Statement

The Attempt at a Solution

I think I first need to find  T(e[SUB]2[/SUB])=? and T(e[SUB]2[/SUB])=? and then combine those into a matrix. 

I am having trouble starting to solve for T(e[SUB]1[/SUB]) and T(e[SUB]2[/SUB])

so far I have   [1] = alpha [1]  + beta [3]
                [0]         [2]         [4]
                                     

I am trying to solve for alpha and beta to find e1


for e2 I have

so far I have   [0] = alpha [1]  + beta [3]
                [1]         [2]         [4]
                           
I am trying to solve for alpha and beta to find e2.

Once I solve these for T(e[SUB]1[/SUB]) & T(e[SUB]2[/SUB]) do I just combine the vectors for the standard matrix?
 
Guidance would be great.

Thank You.

 

[Mark44]

T(<1, 2> = T(1<1, 0>) + 2<0, 1>) and T(<3, 4> = T(3<1, 0>) + 4<0, 1>), right?

 

[schmidtc89]

T(<1, 2> = T(1<1, 0>) + 2<0, 1>) and T(<3, 4> = T(3<1, 0>) + 4<0, 1>), right?

Yes the above is right and makes sense.

 

[schmidtc89]

I am not sure if I did it right.

For t(e1) I got <0,1,2>
For t(e2) I got <3/2,1/2,-1/2>

Standard matrix T( e1 e2 ) -----> <0,1,2><3/2,1/2,-1/2>

 

[Mark44]

Yes, this is correct. You can check by multiplying your 3 x 2 matrix with the two vectors <1, 2> and <3, 4> (transposed).

 

[schmidtc89]

Thanks