# Thread: Proofs in linear algebra

1. ## Proofs in linear algebra

I'm in a linear algebra class and I'm quite new to proofs... In my textbook there are quite a bit of questions that says to prove a bunch of properties and whatnot.

For example...

How would I go about proving

(A^T)^p = (A^p)^T

where p is a non-negative integer and A is a square matrix. (T = transposed)

Or

If A and B is a n x n matrix, prove that AB = BA

How do I prove these? Do I provide an example or something or do I have to do something with summation notation?

2. 1) Use the equality: $\displaystyle (A\cdot B)^T=B^T\cdot A^T$

If $\displaystyle B=A$ then $\displaystyle (A^2)^T=(A^T)^2$

In a similar way you can prove the general case.

2) The equality $\displaystyle AB=BA$ is not true for all A and B.

For example, if $\displaystyle A=\begin{pmatrix}1 & 0\\2 & 0\end{pmatrix}, \ B=\begin{pmatrix}1 & -1\\0 & 0\end{pmatrix}$ then

$\displaystyle AB=\begin{pmatrix}1 & -1\\2 & -2\end{pmatrix}, \ BA=\begin{pmatrix}-1 & 0\\0 & 0\end{pmatrix}$