91Ó°ÊÓ

Test the given set of solutions for linear independence. $$\begin{array}{lll} \text { Differential Equation } & \text { Solutions } \\ y^{\prime \prime \prime \prime}+y^{\prime}=0 & \\{2,-1+2 \sin x, 1+\sin x\\} \end{array}$$

Determine if the subset of \(C(-\infty, \infty)\) is a subspace of \(C(-\infty, \infty)\) The set of all odd functions: \(f(-x)=-f(x)\)

Find a basis for, and the dimension of, the solution space of \(A \mathbf{x}=\mathbf{0}\) $$A=\left[\begin{array}{rr}2 & -1 \\ -6 & 3\end{array}\right]$$

Determine whether the set \(S\) spans \(R^{3}\). If the set does not span \(R^{3}\), give a geometric description of the subspace that it does span. $$S=\\{(1,0,3),(2,0,-1),(4,0,5),(2,0,6)\\}$$

Use a graphing utility or computer software program with matrix capabilities to find the transition matrix from \(B\) to \(B^{\prime}\) $$\begin{array}{l}B=\\{(2,-1,4),(0,2,1),(-3,2,1)\\} \\\B^{\prime}=\\{(1,0,0),(0,1,0),(0,0,1)\\}\end{array}$$

Let \(\mathbf{u}=(1,2,3), \mathbf{v}=(2,2,-1),\) and \(\mathbf{w}=(4,0,-4)\). $$\text { Find } 5 \mathbf{u}-3 \mathbf{v}-\frac{1}{2} \mathbf{w}$$

Determine if the subset of \(C(-\infty, \infty)\) is a subspace of \(C(-\infty, \infty)\) The set of all constant functions: \(f(x)=c\)

Determine whether the set, together with the indicated operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails. The set \(\left\\{\left(x, \frac{1}{2} x\right): x \text { is a real number }\right\\}\) with the standard operations

Test the given set of solutions for linear independence. $$\begin{array}{lll} \text { Differential Equation } & \text { Solutions } \\ y^{\prime \prime \prime \prime}+3 y^{\prime \prime}+3 y^{\prime}+y=0 & \left\\{e^{-x}, x e^{-x}, x^{2} e^{-x}\right\\} \end{array}$$

Test the given set of solutions for linear independence. $$\begin{array}{lll} \text { Differential Equation } & \text { Solutions } \\ y^{\prime \prime \prime}+3 y^{\prime \prime}+3 y^{\prime}+y=0 & \left\\{e^{-x}, x e^{-x}, e^{-x}+x e^{-x}\right\\} \end{array}$$