matrix with respect to bases given

Question

f is a linear transformation from P₂ to R³ whose standard matrix is

                                                                                                 A=  1     0   −1

                                                                                                      −1    1     2

                                                                                                        1    1     1

a) Evaluate f(1 + 2x + x²).

 

b) find the matrix of f wth respect to the bases B₁ = {(1 + x),  (1 − x + x²⁾,  1} of P₂ and B₂ = {(1,1,1), (0,1,1), (1,0,1)} of R³. 

Answer 1

(a) f(1 + 2x + x²) = 3x + 4x². This was found by multiplying A by the vector (1,2,1) to obtain the vector (0,3,4).

 

(b) We have to compute

 

ψ^-1Aφ

 

where φ takes a vector that is in terms of B₁ and puts it into the form of the standard basis of P₂, and where ψ takes a vector that is in terms of B₂ and puts it into the standard basis of R³. That is why we are using the inverse of ψ, because we have to do the opposite: take it from the standard basis of R³ back into B₂.

 

To find the matrix of φ, note

 

     | 1 |      | 1 |

φ  | 0 |  =  | 1 |

     | 0 |      | 0 |

so its matrix has the vector on the right as its first column. Likewise its second and third columns can be seen. So we have the matrix of φ is

 

| 1  1  1 |

| 1 -1  0 |

| 1  0  0 |.

 

Likewise the matrix of ψ is

 

| 1  0  1 |

| 1  1  0 |

| 1  1  1 |

 

and its inverse, the matrix of ψ^-1, can be computed to be

 

| 1  1  -1 |

| -1 0   1 |

| 0  -1  1 |.

 

(If you are unsure how to compute the inverse of a matrix, try looking it up on Purple Math.)

 

Now think what we are doing in the initial formula, reading from right to left. We are taking a vector that is in the basis B₁ and converting it to the standard basis by applying φ. Then we are applying the transformation f by using the standard matrix A, then we are converting something in standard basis into B₂ by using ψ^-1.

 

To find your answer, multiply together the three matrices.