91Ó°ÊÓ

Determine whether the set \(S\) is linearly independent or linearly dependent. $$S=\\{(-2,2),(3,5)\\}$$

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

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

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

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 of all \(2 \times 2\) matrices of the form $$\left[\begin{array}{ll}a & b \\ c & 0\end{array}\right]$$ with the standard operations

Determine if the subset of \(C(-\infty, \infty)\) is a subspace of \(C(-\infty, \infty)\) The set of all functions such that \(f(0)=1\)

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 of all \(2 \times 2\) matrices of the form $$\left[\begin{array}{ll}a & b \\ c & 1\end{array}\right]$$ with the standard operations

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

Find a basis for, and the dimension of, the solution space of \(A \mathbf{x}=\mathbf{0}\) $$A=\left[\begin{array}{lll}1 & 4 & 2\end{array}\right]$$

Explain why \(S\) is not a basis for \(P_{2}\) $$S=\left\\{6 x-3,3 x^{2}, 1-2 x-x^{2}\right\\}$$