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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

Attachment 26266

here is the work i did

Attachment 26267

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

Please help

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

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'}[b_{1}]_{B'}.

so let's find [T(b_{1})]_{B} = [T]_{B}[b_{1}]_{B}, and then figure out how to express this in B'-coordinates.

in B-coordinates, b_{1} 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'} =

$\displaystyle \begin{bmatrix}5&1&4\\2&4&-2\\-2&-1&-1 \end{bmatrix}$.

well, let's see if this checks out.

we should have [T]_{B'} = P^{-1}[T]_{B}P, where:

$\displaystyle P = \begin{bmatrix}1&1&1\\1&1&0\\1&0&0 \end{bmatrix}$

now [T]_{B}P =

$\displaystyle \begin{bmatrix}5&4&1\\7&5&2\\5&1&4 \end{bmatrix}$

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

and P^{-1}[T]_{B}P =

$\displaystyle \begin{bmatrix}0&0&1\\0&1&-1\\1&-1&0 \end{bmatrix} \begin{bmatrix}5&4&1\\7&5&2\\5&1&4 \end{bmatrix} = \begin{bmatrix}5&1&4\\2&4&-2\\-2&-1&-1 \end{bmatrix}$

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

Re: some computation error for Matrix representation with respect to 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