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Lets look at T = R^4 -> R^2, Prove that T is a linear transformation.

where : T$ \begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}= \begin{bmatrix} x + z \\ y + w \end{bmatrix}$

Proof : Let A and B be dummy vectors such as

$A= \begin{bmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \end{bmatrix}$ and $B= \begin{bmatrix} b_1 \\ b_2 \\ b_3 \\ b_4 \end{bmatrix}$

$$ T(cA + B) = \begin{bmatrix} ca1 + ca3 +b1 +b3 \\ ca2 + ca4 +b2 +b4 \end{bmatrix} = \begin{bmatrix} ca1 + ca3 \\ ca2 + ca4 \end{bmatrix} + \begin{bmatrix} b1 + b3 \\ b2 + b4 \end{bmatrix} = c \begin{bmatrix} a1 + a3 \\ a2 + a4 \end{bmatrix} + \begin{bmatrix} b1 + b3 \\ b2 + b4 \end{bmatrix} = cT(A)+T(B) $$

Also, $$ T \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\0 \end{bmatrix} $$ so T is an empty vector Is this a sufficient proof?

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All you need to show is that $T$ satisfies $T(cA+B) =cT(A) +T(B)$ for any vectors $A,B$ in $\mathbb{R}^4$ and any scalar from the field, and $T(0) =0$. It looks like you got it. That should be sufficient proof.

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