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I am having troubles with one of my homework exercises.

The problem states:

Prove that T is a linear transformation, and find bases for both N(T) and R(T). Then compute the nullity and rank of T, and verify the dimension theorem. Finally, use the appropriate theorems in this section to determine whether T is one-to-one or onto.

$T:M$$2\times3$$(F) \to M$$2\times2$$(F)$ defined by

$T\begin{bmatrix}a&b&c\\d&e&f\end{bmatrix} =\begin{bmatrix}2a-b&c+2b\\0&0\end{bmatrix}.$

Showing that this is a linear transformation is where I'm getting hung up. Any help would be appreciated.

Thanks.

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    It is a linear transformation from a vector space of dimension 6 to dimension 2 , so let A and B be matrices show that then T(A + B) = T(A) + T(B) .This just follows from addition of matrices – 2017-02-16

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Let $\phi : M_{2 \times 3}(F) \to F \,\,; \,\, \left[ \begin{matrix} a & b & c \\ d & e & f \end{matrix} \right] \mapsto 2a - b$

Let $\psi : M_{2 \times 3}(F) \to F \,\,; \,\, \left[ \begin{matrix} a & b & c \\ d & e & f \end{matrix} \right] \mapsto c + 2b$

Let $\theta: M_{2 \times 3}(F) \to F \,\,; \,\, \left[ \begin{matrix} a & b & c \\ d & e & f \end{matrix} \right] \mapsto 0$

Can you see why $\phi,\psi,\theta$ are linear transformations? You can use this to show that

$T : M_{2 \times 3}(F) \to M_{2 \times 2}(F) \,\,; \,\, A := \left[ \begin{matrix} a & b & c \\ d & e & f \end{matrix} \right] \mapsto \left[ \begin{matrix} \phi(A) & \psi(A) \\ \theta(A) & \theta(A) \end{matrix} \right]$

is linear as follows: for any scalar $\lambda \in F$ and any matrices $A,B \in M_{2 \times 3}(F)$ we find

$T(\lambda A + B) = \left[ \begin{matrix} \phi(\lambda A + B) & \psi(\lambda A + B) \\ \theta(\lambda A + B) & \theta(\lambda A + B) \end{matrix} \right]$

$= \left[ \begin{matrix} \lambda\phi (A) + \phi(B) & \lambda \psi(A) + \psi(B) \\ \lambda\theta( A) + \theta(B) & \lambda\theta(A) + \theta(B) \end{matrix} \right]$

$ = \lambda \left[ \begin{matrix} \phi (A) & \psi(A) \\ \theta( A) & \theta(A) \end{matrix} \right] = \left[ \begin{matrix} \phi (B) & \psi(B) \\ \theta( B) & \theta(B) \end{matrix} \right] = \lambda T(A) + T(B)$,

using the linearity of $\phi,\psi,\theta$ and the definition of the vector space operations of a space of matrices.

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    I can see where you're going with this, however, in this case I am having trouble understanding how to define $B$. – 2017-02-17
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    To prove $T$ is linear, you need to show that $T( \lambda A + B) = \lambda T(A) + T(B)$ for any scalar $\lambda \in \mathbb{R}$ and any matrices $A,B \in M_{2\times 3}(F)$. All you need to know is that $A,B$ are arbitrary matrices in the domain of $T$; no more specificity is needed because you know that $\phi,\psi,\theta: M_{2 \times 3}(F) \to F$ are linear. I'll update my post. – 2017-02-17
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Hint:

we can represent the given transformation with a matrix as:

$$T(X)= \begin{bmatrix} 2&-1&0&0&0&0\\ 0&2&1&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ \end{bmatrix} \begin{bmatrix} a\\ b\\ c\\ d\\ e\\ f\\ \end{bmatrix} = \begin{bmatrix} 2a-b\\ 2b+c\\ 0\\ 0\\ \end{bmatrix} $$ where the entries of the vectors are the component with respect to the standard basis of $M_{2\times 3}(K)$ and $M_{2\times 2}(K)$.

So you see that $T$ is linear and you can find all its properties.

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    This is more of an intuitive visualization of this transformation for me, but what is my next step? I need to show that addition and scalar multiplication is preserved. So how do I define a $y$ so that $T(x+y)=T(x)+T(y)$ and $T(cx)=cT(x)$? – 2017-02-17
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    Since a matrix always represents a linear transformation, if your transformation is represented by a matrix it is linear. – 2017-02-17