Question

determine whether the following are linear transformation T:M22->M23 ,where B is a fixed 2*3 matrix and T(A)=AB

Answer 1

\[T(c\vec u)=cT(\vec u)\]is easy to prove, but the other one is filling up my paper :/

Answer 2

it seems to work though

Answer 3

\[T(A)=\left[\begin{matrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{matrix}\right]\left[\begin{matrix}b_{11}&b_{12}&b_{13}\\b_{21}&b_{22}&b_{23}\end{matrix}\right]\]\[=\left[\begin{matrix}a_{11}b_{11}+a_{12}b_{21}&a_{11}b_{12}+a_{12}b_{22}&a_{11}b_{13}+a_{11}b_{33}\\a_{21}b_{11}+a_{22}b_{21}&a_{21}b_{12}+a_{22}b_{22}&a_{21}b_{13}+a_{22}b_{33}\end{matrix}\right]\]

Answer 4

okay I'm bored now... do the same thing for \(T(A+B)\) and verify it yourself
the scalar part should be a piece of cake