Denote by $MQ$ the set of 3×3 magic squares and let $MQ_a$ be the set of magic squared with magic sum $a$. The sets $MQ$ and $MQ_0$ are vector spaces over $\Bbb R$ with matrix addition and scalar multiplication.
a) find a linear map $T:\Bbb R^{3×3}\to\Bbb R^8$ such that $\ker(T)$ is precisely the vector space $MQ_0$
b) let $e_{ij}$ be the matrix that has $1$ at entry $ij$ and $0$ elsewhere. What is the matrix of $T$ in the ordered basis $(e_{11}, e_{12}, e_{13}, e_{21}, e_{22}, e_{23}, e_{31}, e_{32}, e_{33})$ for $\Bbb R^{3×3}$ and the standard basis $(e_1,\dots,e_8)$ in $\Bbb R^8$?
For a) I think the linear map take a 3×3 matrix and outputs an 8×1 matrix where each row is the sum of a row, column or diagonal of the original 3×3, of which there are 8 of. Therefore, the $\ker(T)$ should be the vector space of $MQ_0$, as all of the rows, columns and diagonals of the original 3×3 sum to 0
My input is a 3×3 matrix ([x1 x2 x3],[x4 x5 x6],[x7,x8,x9]) reading across the rows. My output is then ([x1+x2+x3},[x4+x5+x6],[x7+x8+x9],[x1+x4+x7],[x2+x5+x8],[x3+x6+x9],[x1+x5+x9],[x3+x5+x7])
For b) I have found a matrix to describe $T$ by writing the columns as $T(e_{11})$ and so on but this forms an 8×9 matrix. I don't see how this can then be multiplied by a 3×3 matrix to get the 8×1 matrix. Perhaps I should rewrite my 3×3 as a 9×1?
Any help is much appreciated thank you!!
Best Answer
Basically a matrix must take as its input a column vector and output a column vector.
Suppose I have a linear transformation whose input is a matrix. Let's say
$$T \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = \begin{pmatrix} a + b \\ c + d \\ a + c \\ b + d \end{pmatrix} $$
I convert that into a matrix equation by picking an ordering on the elements of the matrix. Reasonable choices here are $a,b,c,d$ and $a,c,b,d$ (either reading the matrix by row or by column).
Once the ordering is fixed, we can write this as a matrix equation:
$$ \begin{pmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{pmatrix} \begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix} = \begin{pmatrix} a + b \\ c + d \\ a + c \\ b + d \end{pmatrix}. $$
Your task is to do the same but for a $3\times 3$ matrix and with the ordering given to you: $e_{11}, e_{12}, e_{13}$, etc. It will also depend on how you order the equations in your transformation $T$.