"Can i solve it backward by finding W first then use[W]b' × b' = W to find [W]b'
However i dont understand what u want to find from question18"

Imagine an input vector in 2 non-standard bases as well as the standard basis:

v,     [v]B,     [v]B'

They are all the same point but written in different bases (can think of it as different coordinate systems).

Then imagine you have a linear transformation in the standard basis represented by the matrix A. Working only in the standard basis we would have,

Av = w

Now, we could write the answer in the basis, B':

[w]B'

We could also write the input vector, v, in the other basis, B:

[v]B

Question 18 is asking, what would the matrix be (A*) if we want to do this:

A*[v]B = [w]B'

Remember that v and w are always the same numbers but just written in different bases (coordinates). We are just saying, what if the input vector is one basis, and we want the output in a different one, and neither basis is the standard one.

The thinking is like this:

- I already have a transformation matrix, A, which is in the standard basis.

- Somebody has given me a vector in the basis, B:      [v]B

- I need to transform it and get the answer, but the person wants the answer in a different basis, B':     [w]B'

- Usually I would have to do this:

[v]B    >>>    convert to v (standard basis)    >>>   do transformation Av=w    >>>    convert answer to [w]B'

  but I can also find a matrix, A* that does the job for me.

The question is about getting that matrix A*.

I am not sure what you mean by " [W]b' × b' ", since b' is a basis (set of vectors) so you cannot multiply it by a vector.