1. I.14 Linear Spaces

Condition: to prove that, many \ (m \) forms a subspace in space \ (m_ {m \ times n} \) of all matrices of this size. Find the dimension and build the basis \ (m \). Check that the matrix \ (b \) belongs to \ (m \) and decompose it according to the base in \ (m \). \ Begin {Tabular} {| l | cc |} \ hline \ (\ begin {array} {l} m- \ text {Many matrices \\ \ Text {indicated species} \ End {Array} \) &) &) &) \ multicolumn {2} {| c |} {\ (b \)} \\ \ \ hline \ (\ begin {array} {l} \ text {nizhnnet-worn matrices 3-} \\ \ text {th order with zero afterwards and \\ \\ \\ \\ \\ \ Text {zero sum of the elements} \\ \ Text {side diagonal} \ end {array} \) & \ (\ begin {array} {ccccc} 1 & 0 \\ 4 & 2 & 2 &2 &2 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5EN -3 \ end {Array} \ Right) \) \\ \ HLINE \ END {Tabular}