MathProblemsBank

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\}, \[ B=\left(\begin{array}{ccc} 0 & 1 & -1 \\ 1 & -1 & 0 \\ -1 & 0 & 1 \end{array}\right) \]