91Ó°ÊÓ

Decide (with justification) whether \(S\) is a subspace of \(V\) $$V=C[a, b], S=\left\\{f \in V: \int_{a}^{b} f(x) d x=0\right\\}$$

Let \(S\) be the subspace of \(\mathbb{R}^{3}\) consisting of all solutions to the linear system $$ x-2 y-z=0 $$ Determine a set of vectors that spans \(S .\)

Find the change-of-basis matrix \(P_{C \leftarrow B}\) from the given ordered basis \(B\) to the given ordered basis \(C\) of the vector space \(V.\) $$\begin{aligned}&V=M_{2}(\mathbb{R});\\\&B=\left\\{\left[\begin{array}{rr}1 & 0 \\\\-1 & -2 \end{array}\right],\left[\begin{array}{cc}0 & -1 \\\3 & 0\end{array}\right],\left[\begin{array}{cc} 3 & 5 \\\0 & 0\end{array}\right],\left[\begin{array}{cc}-2 & -4 \\\0 & 0\end{array}\right]\right\\}\\\&C=\left\\{\left[\begin{array}{ll}1 & 1 \\\1 & 1\end{array}\right],\left[\begin{array}{ll}1 & 1 \\\1 & 0\end{array}\right],\left[\begin{array}{ll} 1 & 1 \\\0 & 0\end{array}\right],\left[\begin{array}{ll}1 & 0 \\\0 & 0\end{array}\right]\right\\} \end{aligned}$$.

Verify that the given set of objects together with the usual operations of addition and scalar multiplication is a complex vector space. $$C^{2}.$$

Determine a spanning set for the null space of the given matrix \(A\) The matrix \(A\) defined in Problem 25 in Section \(4.3 .\)

Use the Wronskian to show that the given functions are linearly independent on the given interval \(I\). $$f_{1}(x)=1, f_{2}(x)=x, f_{3}(x)=x^{2}, I=(-\infty, \infty)$$.

Let \(S\) denote the set of all \(4 \times 4\) matrices such that the entries in each row and each column add up to zero. (a) Show that \(S\) is a subspace of \(M_{4}(\mathbb{R})\) (b) Find a basis and the dimension of \(S\) (c) Extend the basis you constructed in part (b) to a basis for \(M_{4}(\mathbb{R})\)

Give a geometric description of the subspace of \(\mathbb{R}^{3}\) spanned by the given set of vectors. \(\left\\{\mathbf{v}_{1}\right\\},\) where \(\mathbf{v}_{1}\) is any nonzero vector in \(\mathbb{R}^{3}\)