Hi,
For this problem I am supposed to find the bases B and C of the source and target vector spaces V and W and write the matrix [T]b,c that represents T with respect to these basis.
I understand that this should be a 1xn matrix as we are moving from a Matrices with dim(nxn) to the space R, dim(1). But I am not sure what this [T]b,c matrix would look like. Any insight would be greatly appreciated!
Best Answer
The dimension of $M_{n\times n}(\mathbb{R})$ is $n^2$. A basis is given by the $n^2$ matrices $e_{ij}$, where $1 \leq i,j \leq n$ and $e_{ij}$ has a single one in position $(i,j)$ and zeroes elsewhere. Therefore the matrix of $T$ has size $1 \times n^2$. For this problem it is more convenient to order the basis elements of $M_{n\times n}(\mathbb{R})$ as $(e_{11},e_{22},\ldots,e_{nn},e_{12},e_{13},\ldots)$. (I.e. place the $n$ diagonal basis elements before the rest.) Then since $T(e_{ij}) = \operatorname{tr}(e_{ij}) = 1 \text{ or } 0$ depending on whether $i=j$ or not, we find that the matrix of $T$ is $(1,1,\ldots,1,0,\ldots,0)$. (There are $1$s $n$ times.)