91Ó°ÊÓ

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. \(S=\\{(-3,5)\\}\)

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

Find the coordinate matrix of \(\mathbf{x}\) in \(R^{n}\) relative to the basis \(B^{\prime}\) $$B^{\prime}=\left\\{\left(\frac{3}{2}, 4,1\right),\left(\frac{3}{4}, \frac{5}{2}, 0\right),\left(1, \frac{1}{2}, 2\right)\right\\}, \mathbf{x}=\left(3,-\frac{1}{2}, 8\right)$$

In Exercises \(13-16\) find a basis for the subspace of \(R^{3}\) spanned by \(S\). \(S=\\{(2,3,-1),(1,3,-9),(0,1,5)\\}\)

Find the Wronskian for the set of functions. $$ \left\\{e^{3 x}, \sin 2 x\right\\} $$

Find a basis for the subspace of \(R^{3}\) spanned by \(S\). $$ S=\\{(2,3,-1),(1,3,-9),(0,1,5)\\} $$

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. \(S=\\{(1,1)\\}\)

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

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. \(S=\\{(-1,2),(2,-4)\\}\)

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