[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.

Seems i have my whole concept of how to go about transformations misconstrued. Please help. Any comments/suggestions would be greatly appreciated. Thank you in advance and good day. Cheers!)

 

[jambaugh]

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.
Then note that these transformations can be composed by multiplying the transformation matrices.
So Transform From B to Standard Basis and then to B' (and then the reverse).

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.