Question

Show T is a linear transformation

Answer 1

To show something is a linear transformation you need to show 2 things.
1. T(M+N)=T(M)+T(N) for M,N in V
2. T(aM)=aT(M) for a in the field of scalars for V.
Which of these do you need help showing?

Answer 2

The first one would be helpful. I'm just not really sure how to use the definition of T provided...

Answer 3

Well from the definition of T, T(M+N)=P(M+N)Q. The using the distributive property twice we get P(M+N)Q=(PM+PN)Q=PMQ+PNQ, which is just T(M)+T(N) from the definition of T.

Did that make sense?

Answer 4

Yes. My only confusion on what M and N are....

Answer 5

M and N are general elements of V. In other words, they're nxn matrices.

Answer 6

Ohhhhhhh!!!! Makes way more sense now!! Thank you so much!

Answer 7

You're welcome.