Thread: some computation error for Matrix representation with respect to B'

Please some one Help me, I have no idea what to do on part (b) where its asking to find the matrix representation with respect to B'

Here is the original Question

here is the work i did


but when i did the check in step d i found out that my work is wrong

Please help

suppose we want to find out how T transforms vectors in B'-coordinates.

well, all we know at the moment, is what T does to B-coordinates (the standard ones).

but whatever [T]_B' is, we know that the first column of [T]_B' is:

[T]_B'[b₁]_B'.

so let's find [T(b₁)]_B = [T]_B[b₁]_B, and then figure out how to express this in B'-coordinates.

in B-coordinates, b₁ is (1,1,1). and T(1,1,1) = (5,7,5). since this is 5(1,1,1) + 2(1,1,0) - 2(1,0,0), in B'-coordinates, this is [5,2,-2]_B'.

ok, now let's do the same with the 2nd B' basis vector (1,1,0). computing T(1,1,0) = (4,5,1) = 1(1,1,1) + 4(1,1,0) - 1(1,0,0), so this is [1,4,-1]_B'.

finally, we have T(0,0,1) = (1,2,4) = 4(1,1,1) - 2(1,1,0) - 1(1,0,0), so this is [4,-2,-1]_B'.

therefore, it would seem that [T]_B' =

.

well, let's see if this checks out.

we should have [T]_B' = P^-1[T]_BP, where:



now [T]_BP =



(these numbers should look familiar, they are the images of the B'-basis under T in the standard basis).

and P^-1[T]_BP =



i have NO idea what you are doing with the row-reduction in part (b).

Thank you so much, I have basically learned this lesson completely for your detailed worked out solution, and I was able to solve the rest of questions on the lesson, apparently I have a complete misunderstanding
again thank you sooooo much for the work put it in this solution