91Ó°ÊÓ

Determine whether \(\mathbf{b}\) is in the column space of \(A\). If it is, write \(\mathbf{b}\) as a linerr combination of the column yectors of \(A\) $$A=\left[\begin{array}{rrr}1 & 3 & 2 \\ -1 & 1 & 2 \\ 0 & 1 & 1\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l}1 \\ 1 \\\ 0\end{array}\right]$$

Identify and sketch the graph. $$x^{2}+4 x+6 y-2=0$$

Use a graphing utility or computer software program with matrix capabilities to write \(\mathbf{v}\) as a linear combination of \(\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}, \mathbf{u}_{4},\) and \(\mathbf{u}_{5},\) or of \(\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}, \mathbf{u}_{4}, \mathbf{u}_{5},\) and \(\mathbf{u}_{6} .\) Then verify your solution. $$\begin{aligned} \mathbf{u}_{1} &=(1,2,-3,4,-1) \\ \mathbf{u}_{2} &=(1,2,0,2,1) \\ \mathbf{u}_{3} &=(0,1,1,1,-4) \\ \mathbf{u}_{4} &=(2,1,-1,2,1) \\ \mathbf{u}_{5} &=(0,2,2,-1,-1) \\ \mathbf{v} &=(5,3,-11,11,9) \end{aligned}$$

Determine whether \(S\) is a basis for \(R^{3}\). If it is. write \(\mathbf{u}=(8,3,8)\) as a linear combination of the vectors in \(S\) $$S=\\{(0,0,0),(1,3,4),(6,1,-2)\\}$$

Explain why the row vectors of a \(4 \times 3\) matrix form a linearly dependent set. (Assume all matrix entries are distinct.)

Use a graphing utility or computer software program with matrix capabilities to write \(\mathbf{v}\) as a linear combination of \(\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}, \mathbf{u}_{4},\) and \(\mathbf{u}_{5},\) or of \(\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}, \mathbf{u}_{4}, \mathbf{u}_{5},\) and \(\mathbf{u}_{6} .\) Then verify your solution. $$\begin{aligned} \mathbf{u}_{1} &=(1,1,-1,2,1) \\ \mathbf{u}_{2} &=(2,1,2,-1,1) \\ \mathbf{u}_{3} &=(1,2,0,1,2) \\ \mathbf{u}_{4} &=(0,2,0,1,-4) \\ \mathbf{u}_{5} &=(1,1,2,-1,2) \\ \mathbf{v} &=(5,8,7,-2,4) \end{aligned}$$

Determine whether the sets \(S_{1}\) and \(S_{2}\) span the same subspace of \(R^{3}\) $$\begin{array}{l}S_{1}=\\{(0,0,1),(0,1,1),(2,1,1)\\} \\\S_{2}=\\{(1,1,1),(1,1,2),(2,1,1)\\}\end{array}$$

Identify and sketch the graph. $$y^{2}+8 x+6 y+25=0$$

Explain why the column vectors of a \(3 \times 4\) matrix form a linearly dependent set. (Assume all matrix entries are distinct.)

Determine whether \(S\) is a basis for \(R^{3}\). If it is. write \(\mathbf{u}=(8,3,8)\) as a linear combination of the vectors in \(S\) $$S=\\{(1,0,1),(0,0,0),(0,1,0)\\}$$