1.i.14 Linear spaces

Condition: Prove that the set \( M \) forms a subspace in the space \( M_{m \times n} \) of all matrices of a given size. Find the dimension and construct the basis \(M\). Check that the matrix \( B \) belongs to \( M \) and expand it with respect to the basis in \( M \). \begin{tabular}{|l|cc|} \hline \( \begin{array}{l}M-\text { set of matrices } \\ \text { of the specified type }\end{array} \) & \multicolumn{2}{|c|}{\( B \)} \\ \hline \( \begin{array}{l}\text { Lower triangular matrices of 3- } \\ \text { of the first order with zero trace and } \\ \text { zero sum of elements along } \\ \text { secondary diagonal }\end{array} \) & \( \left(\begin{array}{ccc}1 & 0 & 0 \\ 4 & 2 & 0 \\ -2 & 5 & -3\end{array}\right) \) \\ \hline \end{tabular}