Finding a Linear Transformation with specific domail and range

[apdixon]

Hey, i have an assignment in MATLAB class which is
Let L be a linear transformation such that
L(1)=(2 -1)'
L(1-x)=(1 0)'
L(1+x^2)=(1 1)'
L(1+x^3)=(1 2)'
Determine a matrix in domain such that with the canonical in range, the matrix that represents L has two null columns.

I don't know exactly what they're asking me to find. I found L by solving LP=A, being A the transformed matrix [2 1 1 1;-1 0 1 2], and P the matrix P=[1 1 1 1;0 -1 0 0;0 0 1 0;0 0 0 1]. But it doesn't have two null columns, and it was too easy to find to be true. Pleaaaase help!

 

[vela]

You found the matrix for L where the basis for the domain is {1, x, x², x³} (which is reflected in how you constructed P). If you use a different basis for the domain, you'll get a different matrix that represents the linear transformation L. The problem is asking you to find a basis for the domain so that the matrix has two null columns.

 

[apdixon]

Thank you! I found the Ker for A=[2 1 1 1; -1 0 1 2], then constructed a matrix F with the Ker in the first two columns and (0100)' and (1000)' in the next two so that i had a matrix of 4x4 and then multiplied A*F. I found L=
0 0 1 2
0 0 0 -1 .