# [Linear Algebra] Matrix Transformations

SpaceMonkeyCaln
Homework Statement:
https://prnt.sc/riq7m0
Relevant Equations:
Gauss Jordan
Evening,

The reason for this post is because as the title suggests, I have a question concerning matrix transformation. These are essentially test prep problems and I am quite stuck to be honest.

Here are the [questions](https://prnt.sc/riq7m0) and here are the [answers](https://prnt.sc/riq8b6).

For problem **#14**, pretty simple. [Here](https://prnt.sc/riqb0v) is my solution.

Now, for problem **#15a**, I'm confused. The solution states and solves for P^-1 . How and why? Shouldn't #14 and #15a share the same answer of P given i am simply asked to find the transition matrix from the basis B to the basis B'? And even then, i didn't find the given answer to be the inverse of P so how exactly can i go about solving this.?

As for problem **#15b**, i tried doing the reverse of problem #14 as i now am asked to solve BB' rather than B'B. So, in solving BB', i got an answer that doesn't resemble the given answer in the slightest. Not sure what I'm doing wrong.

Consider how you would map the standard basis $\left\{ \left[\begin{array}{c} 1\\0 \end{array}\right], \left[\begin{array}{c} 0\\1 \end{array}\right]\right\}$to either B or B' and then back again.
Secondly note that left multiplication by a square matrix transforms column vectors and columns of a general matrix in the same way. So you can transform both basis vectors by forming their columns into a 2x2 matrix and left multiplying by your transformation matrix. $T\cdot B_1 = B_2$ as a product of two square matrices forming a square matrix.