Finding matrix relative to different bases

[feinm1]

Homework Statement

Let B = {b₁, b₂, b₃} be a basis for a vector space V and T : V -> R² be a linear transformation with the property that

T(x₁b₁ + x₂b₂ + x₃b₃) =

2x₁ - 4x₂ + 5x₃
0x₁ - 1x₂ + 3x₃

Find the matrix for T relative to B and the standard basis for R².

Homework Equations

[T(x)]_C = M[x]_B

Where M = [[T[b₁)]_C [T(b₂)]_C ... [[T(b_n)]_C

This matrix M is known as the matrix for T relative to the Bases B and C.

The Attempt at a Solution

[T(x)]_C =

2x₁ - 4x₂ + 5x₃
0x₁ - 1x₂ + 3x₃

If [x_B] =

|x₁|
|x₂|
|x₃|

Then M =

|2 -4 5|
|0 -1 3|

Is this correct?

 

[Klockan3]

Is this correct?

Yes, this is just a problem to check if you have understood the concept since they basically gave you the solution. A good way to do these is to check what the standard basis vectors would become, e1 becomes the first column, e2 the second etc. Also always make sure that you are transforming them in the right direction, it is quite easy to mess that up if you don't think while doing this.