There exists a linear transformation that maps a $2×2$ matrix to $R^3$ vector?

Question

My question is about linear transformations, Is there a linear transformation that maps a $2\times2$ matrix to a vector in $R^3$, like:

$T:M_{2}[R]\to{R^3}$

Such that:

$T\Biggl($$\begin{bmatrix}1&0\\0&0\end{bmatrix}\Biggr)$ $ = T\Biggl($$\begin{bmatrix}0&1\\0&0\end{bmatrix}\Biggr)$ $ = \bigl(1,2,3\bigr)$ $?$

I spent alot of time thinking about a possible solution without success.

And in general, Is it possible to define any linear transformation $T$ that maps a $2\times2$ matrix to a vector in $R^3$ such that: $T:M_{2}[R]\to{R^3}$ $?$ Since for every linear transformation there is a matrix $A$ such that: $T(x) = Ax$, what the dimensions of that matrix would be?

Thanks for help!!

Answer 1

Given any vector spaces $V$ and $W$ you can define a linear transformation $T:V\rightarrow W$ simply by declaring $$ T(e_i)=w_i $$ where $e_i$ runs through a basis of $V$ and the $w_i$ are an arbitrary choice of vectors in $W$.

Therefore the answer to your first question is yes.

To answer your second question you need to remember how $T$ is coded into the matrix $A$.The columns of $A$ are the coordinates of the images of a basis of $V$ in term of a prefixed basis of $W$. Thus $A$ has $m=\dim V$ columns and $n=\dim W$ rows.

Answer 2

You can simply define, for example, $$ T\begin{pmatrix} x & y \\ z & w \end{pmatrix} = (x+y,2x+2y,3x+3y) $$ and verify directly that function defined in that ways satisfies the conditions for being a linear transformation.

(For the matrix representation of $T$, see trancelocation's answer).

Answer 3

To get a matrix representation of $T$ you need to indicate a basis $B=\{b_1,b_2,b_3,b_4\}$ in $M_{2}[\mathbb{R}]$. For example

• $b_1 = \begin{bmatrix}1&0\\0&0\end{bmatrix}$, $b_2 = \begin{bmatrix}0&1\\0&0\end{bmatrix}$, $b_3 = \begin{bmatrix}0&0\\1&0\end{bmatrix}$, $b_4 = \begin{bmatrix}0&0\\0&1\end{bmatrix}$

Then, a possible matrix representation for just projecting onto the subspace spanned by $(1,2,3)^t$ is

$$A_T = \begin{bmatrix}1&1&0&0\\2&2&0&0\\3&3&0&0\end{bmatrix}$$

Answer 4

$$T\Biggl(\begin{bmatrix}a&b\\c&d\end{bmatrix}\Biggr)=(a,b,c) $$

is such a linear transformation.