91Ó°ÊÓ

Find a homogeneous system whose solution space is spanned by the following sets of three vectors: (a) \((1,-2,0,3,-1),(2,-3,2,5,-3),(1,-2,1,2,-2)\); (b) \((1,1,2,1,1),(1,2,1,4,3),(3,5,4,9,7)\).

Determine whether each of the following is a basis of the vector space \(\mathbf{P}_{n}(t)\) : (a) \(\left\\{1, \quad 1+t, \quad 1+t+t^{2}, \quad 1+t+t^{2}+t^{3}, \quad \ldots, \quad 1+t+t^{2}+\cdots+t^{n-1}+t^{n}\right\\}\) (b) \(\quad\left\\{\begin{array}{lllll}1+t, & t+t^{2}, & t^{2}+t^{3}, & \ldots, & t^{n-2}+t^{n-1}, & \left.t^{n-1}+t^{n}\right\\} .\end{array}\right.\)

For \(k=1,2, \ldots, 5\), find the number \(n_{k}\) of linearly independent subsets consisting of \(k\) columns for each of the following matrices: (a) \(A=\left[\begin{array}{lllll}1 & 1 & 0 & 2 & 3 \\ 1 & 2 & 0 & 2 & 5 \\ 1 & 3 & 0 & 2 & 7\end{array}\right]\) (b) \(\quad B=\left[\begin{array}{lllll}1 & 2 & 1 & 0 & 2 \\ 1 & 2 & 3 & 0 & 4 \\\ 1 & 1 & 5 & 0 & 6\end{array}\right]\)

Show that if any row is deleted from a matrix in echelon (respectively, row canonical) form, then the. resulting matrix is still in echelon (respectively, row canonical) form.

Suppose \(U\) and \(W\) are subspaces of \(V\) such that \(\operatorname{dim} U=4, \operatorname{dim} W=5\), and \(\operatorname{dim} V=7\). Find the possible dimensions of \(U \cap W\).