Proving Matrix Transformation Property

[RJLiberator]

Homework Statement

Let A and B be n x m matrices, and λ and μ be real numbers. Prove that:
$(λA+μB)^T = λA^T+μB^t$

Homework Equations

:/

The Attempt at a Solution

I'm struggling to start here.

If there was no λ and μ, I think I'd be able to reasonably solve this. How do I show that these real numbers don't matter in this exchange?

 

[LCKurtz]

Homework Statement

Let A and B be n x m matrices, and λ and μ be real numbers. Prove that:
$(λA+μB)^T = λA^T+μB^t$

Homework Equations

:/

The Attempt at a Solution

I'm struggling to start here.

If there was no λ and μ, I think I'd be able to reasonably solve this. How do I show that these real numbers don't matter in this exchange?

Let $A = (a_{ij})$ and $B=(b_{ij})$ and show that the corresponding elements on the two sides are equal.

 

[RUber]

If there was no λ and μ, I think I'd be able to reasonably solve this. How do I show that these real numbers don't matter in this exchange?

There is really no difference between showing it with a scalar multiple and without the scalar multiple, since the scalars distribute to every term in the matrix. You could just as easily let $A_\lambda = \lambda A$ and $B_\mu = \mu B$ and show what you say you could show for the new matrices. Then the last step is to show that you can pull the scalars back out...since they are distributed to each term, they will not affect the solution.

 

[RJLiberator]

Lc and Ruber, thanks for the words of advice. What you stated resonates well with me. I will be working on this problem in the next 24 hours and I feel confident about it. I will post back with my update. :)

 

[RJLiberator]

Got it.
Using the transpose definition I make the original statement
(λA+μB)_ji
and then by previous axioms/operations it flows smoothly to λ(A_ji)+μ(B_ji)

A use of the transpose definition again
(λA^T+μB^T)

And we have shown what was needed to be shown.
:) Thank you.