91Ó°ÊÓ

determine whether the set, together with the standard 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 third-degree polynomials

Find a basis for the subspace of \(R^{3}\) spanned by \(S\). $$ S=\\{(4,4,8),(1,1,2),(1,1,1)\\} $$

Find the coordinate matrix of \(\mathbf{x}\) in \(R^{n}\) relative to the basis \(B^{\prime}\) $$\begin{aligned}&B^{\prime}=\\{(4,3,3),(-11,0,11),(0,9,2)\\}\\\&\mathbf{x}=(11,18,-7)\end{aligned}$$

\(W\) is not a subspace of the vector space. Verify this by giving a specific example that violates the test for a vector subspace (Theorem 4.5). \(W\) is the set of all matrices in \(M_{3,3}\) of the form \(\left[\begin{array}{lll}1 & a & b \\ c & 1 & d \\ e & f & 0\end{array}\right]\)

Find the vector \(\mathbf{v}\) and illustrate the specified vector operations geometrically, where \(\mathbf{u}=(-2,3)\) and \(w=(-3,-2)\). $$\mathbf{v}=\frac{1}{2}(3 \mathbf{u}+\mathbf{w})$$

In Exercises \(13-16\) find a basis for the subspace of \(R^{3}\) spanned by \(S\). \(S=\\{(4,4,8),(1,1,2),(1,1,1)\\}\)

Find the Wronskian for the set of functions. $$ \left\\{e^{x^{2}}, e^{-x^{2}}\right\\} $$

In Exercises \(13-16\) find a basis for the subspace of \(R^{3}\) spanned by \(S\). \(S=\\{(1,2,2),(-1,0,0),(1,1,1)\\}\)

Find the coordinate matrix of \(\mathbf{x}\) in \(R^{n}\) relative to the basis \(B^{\prime}\) $$\begin{array}{l}B^{\prime}=\\{(9,-3,15,4),(3,0,0,1),(0,-5,6,8) \\\\(3,-4,2,-3)\\} \\\x=(0,-20,7,15)\end{array}$$

Find a basis for the subspace of \(R^{3}\) spanned by \(S\). $$ S=\\{(1,2,2),(-1,0,0),(1,1,1)\\} $$