Help Solving Linear Transformation Problem in R^2

[akimbo]

Hey i got a problem here but still without correction so if you guys can help me , thanks in advance I'm stuck there

We have L : P -> R^2
L is a linear transformation with :

$$B = \left\{1-x^{2},2x,1+2x+3x^{2} \right\} \; and \; B' = \begin{Bmatrix} \begin{bmatrix} 1\\-1 \end{bmatrix} \begin{bmatrix} 2\\0 \end{bmatrix} \end{Bmatrix} as \; [L]^{B'}_{B} = \begin{bmatrix} 2 &-1 &3 \\ 3&1 & 0 \end{bmatrix}$$

I have to find
1/ the transfer matrix from B' to the canonical basis of R^2
also
2/ the transfer matrix from the canonical basis ( 1 , x , x² )of P in the B basis
3/ find matrix L in the canonical basis of P and R^2

 

[jvicens]

Thank you for reaching out for help. It seems like you are working on linear transformations and transfer matrices. Let me try to explain the steps you need to take to answer the questions you have.

1. To find the transfer matrix from B' to the canonical basis of R^2, you need to first find the inverse of the matrix [L]^{B'}_{B}. This can be done by using the Gauss-Jordan elimination method or by using the inverse matrix formula. Once you have the inverse, you can simply multiply it with the matrix [L]^{B'}_{B}. The resulting matrix will be the transfer matrix from B' to the canonical basis of R^2.

2. To find the transfer matrix from the canonical basis of P (1, x, x^2) to the B basis, you need to first find the matrix ^{B}_{P} which represents the identity transformation from P to B. This can be done by writing the vectors in B as linear combinations of the vectors in the canonical basis of P. Then, you can simply multiply ^{B}_{P} with the matrix [L]^{B}_{B'} to get the transfer matrix from the canonical basis of P to the B basis.

3. To find the matrix L in the canonical basis of P and R^2, you can use the transfer matrices you have found in the previous steps. The matrix L in the canonical basis of P and R^2 can be written as [L]^{B}_{P} [L]^{B'}_{B} ^{B}_{P}.

I hope this helps you solve the problem. Good luck!