Exam 4: Vector Spaces

Find an explicit description of the null space of matrix A by listing vectors that span the null space. - $A = \left[ \begin{array} { r r r r } 1 & - 2 & - 2 & - 2 \\0 & 1 & 3 & 4\end{array} \right]$

Find a matrix A such that W = Col A. - $A = \left[ \begin{array} { r r } 1 & - 2 \\0 & 6 \\- 4 & 5 \\- 1 & - 3 \\3 & - 5\end{array} \right]$

Find the steady-state probability vector for the stochastic matrix P. - $\mathrm { P } = \left[ \begin{array} { l l } 0.1 & 0.5 \\0.9 & 0.5\end{array} \right]$

Find a basis for the column space of the matrix. -Determine which of the following statements is true. A: If $\mathrm { V }$ is a 4 -dimensional vector space, then any set of exactly 4 elements in $V$ is automatically a basis for $V$ . B: If there exists a set $\left\{ \mathbf { v } _ { 1 } , \ldots \ldots , \mathbf { v } _ { 5 } \right\}$ that spans $V$ , then $\operatorname { dim } V = 5$ . $\mathrm { C }$ : If $\mathrm { H }$ is a subspace of a finite-dimensional vector space $\mathrm { V }$ , then $\operatorname { dim } \mathrm { H } \leq \operatorname { dim } \mathrm { V }$ .

Find the new coordinate vector for the vector x after performing the specified change of basis. - $y \mathrm { k } + 3 - 4 \mathrm { y } _ { \mathrm { k } } + 2 - 4 \mathrm { y } \mathrm { k } + 1 + 16 \mathrm { y } _ { \mathrm { k } } = 0$ for all $\mathrm { k }$

Determine whether {v1, v2, v3} is a basis for $\mathbf { b } _ { 1 } = \left[ \begin{array} { l } 1 \\ 1 \end{array} \right] , \mathbf { b } _ { 2 } = \left[ \begin{array} { r } 1 \\ - 1 \end{array} \right] , \mathbf { x } = \left[ \begin{array} { r } 3 \\ - 5 \end{array} \right]$ , and $B = \left\{ \mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } \right\}$ - $\mathbf { b } _ { 1 } = \left[ \begin{array} { r } 3 \\ 2 \\ - 3 \end{array} \right] , \mathbf { b } _ { 2 } = \left[ \begin{array} { r } 5 \\ - 3 \\ - 1 \end{array} \right] , \mathbf { x } = \left[ \begin{array} { r } - 16 \\ 2 \\ 8 \end{array} \right]$ , and $B = \left\{ \mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } \right\}$

Determine if the vector u is in the column space of matrix A and whether it is in the null space of A. - $\mathbf { v } _ { 1 } = \left[ \begin{array} { r } 1 \\- 3 \\3\end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { r } - 3 \\8 \\8\end{array} \right] , \mathbf { v } _ { 3 } = \left[ \begin{array} { r } 2 \\- 2 \\- 7\end{array} \right]$