[Math] Verifying a Linear transformation from M 2,2 -> R

linear algebralinear-transformations

I'm trying to verify whether or not the function: 

T:M(2,2) -> R, T(A) = a + b - c + d, where A =

\begin{bmatrix}a&b\\c&d\end{bmatrix}

From what I understand, T(v) = Av is a linear transformation and thus I need to prove this from my given function. However I don't know how to go about doing this. I also know that there are two properties of linear transformations:

T(u+v) = T(u) + T(v)
T(cu) = cT(u)

But again, I'm lost how to verify this "linear transformation" from matrix to R.

Best Answer

$U = \begin{bmatrix}u_{11}&u_{12}\\u_{21}&u_{22}\end{bmatrix}$

$V = \begin{bmatrix}v_{11}&v_{12}\\v_{21}&v_{22}\end{bmatrix}$

$U+V = \begin{bmatrix}u_{11}+v_{11}&u_{12}+v_{12}\\u_{21}+v_{21}&u_{22}+v_{22}\end{bmatrix}$

$T(U+V) = u_{11}+v_{11}+u_{12}+v_{12}-u_{21}-v_{21}+u_{22}+v_{22}$ = T(U)+T(V)

Now show that $T(cU) = cT(U)$

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