
1.9.2 Linear spaces
Problem:
Prove that the set of matrices \( M \) is a subspace in the space of all matrices of a given size. Construct a basis and find the dimension of the subspace \( M \). Check that the matrix \( \mathrm{B} \) belongs to \( \mathrm{M} \) and decompose it according to the found basis.
\( M=\left\{A \in M_{3 \times 3} \mid A=A^{T}\right. \) (symmetrical), the sums of the elements in the columns are the same, sums of elements in rows alternate\},
\[
B=\left(\begin{array}{ccc}
0 & 1 & -1 \\
1 & -1 & 0 \\
-1 & 0 & 1
\end{array}\right)
\]