91Ó°ÊÓ

In Exercises \(17-20\), find a basis for the subspace of \(R^{4}\) spanned by \(S\). \(\begin{aligned} S=\\{&(6,-3,6,34),(3,-2,3,19),(8,3,-9,6) \\ &(-2,0,6,-5)\\} \end{aligned}\)

Find the vector \(\mathbf{v}\) and illustrate the specified vector operations geometrically, where \(\mathbf{u}=(-2,3)\) and \(w=(-3,-2)\). For the vector \(\mathbf{v}=(3,-2),\) sketch (a) \(4 \mathbf{v},(\mathbf{b})-\frac{1}{2} \mathbf{v},\) and (c) \(0 v\)

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

Find the Wronskian for the set of functions. $$ \\{x,-\sin x, \cos x\\} $$

Find the transition matrix from \(B\) to \(B^{\prime}\) $$B=\\{(1,0),(0,1)\\}, B^{\prime}=\\{(1,1),(5,6)\\}$$

Find a basis for the subspace of \(R^{4}\) spanned by \(S\). $$ \begin{aligned} S=\\{&(6,-3,6,34),(3,-2,3,19),(8,3,-9,6) \\ &(-2,0,6,-5)\\} \end{aligned} $$

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 first-degree polynomial functions \(a x+b\) \(a, b \neq 0,\) whose graphs do not pass through the origin

\(W\) 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 \(M_{n, n}\) such that \(A^{2}=A\)

Determine whether the set \(S\) spans \(R^{3} .\) If the set does not span \(R^{3},\) then give a geometric description of the subspace that it does span. \(S=\\{(4,7,3),(-1,2,6),(2,-3,5)\\}\)

Find the transition matrix from \(B\) to \(B^{\prime}\) $$B=\\{(2,4),(-1,3)\\}, B^{\prime}=\\{(1,0),(0,1)\\}$$