Exam 4: Vector Spaces

Find the general solution of the difference equation. -Given that the signal $y _ { k } = 3 k - 5$ is a solution of the given difference equation, find a description of all solutions of the equation. $y _ { k + 2 } - 6 y _ { k + 1 } + 5 y _ { k } = - 12$ for all $k$

Find the steady-state probability vector for the stochastic matrix P. -Suppose that demographic studies show that each year about 8% of a cityʹs population moves to the suburbs (and 92% stays in the city), while 4% of the suburban population moves to the city (and 96% remains in the suburbs). In the year 2000, 64.1% of the population of the region lived in The city and 35.9% lived in the suburbs. What percentage of the population of the region would Eventually live in the city if the migration probabilities were to remain constant over many years? For simplicity, ignore other influences on the population such as births, deaths, and migration into And out of the region.

Find an explicit description of the null space of matrix A by listing vectors that span the null space. - $\mathbf { u } = \left[ \begin{array} { r } - 1 \\- 2 \\1\end{array} \right] , \mathrm { A } = \left[ \begin{array} { r r r } - 2 & - 3 & - 8 \\- 3 & - 1 & - 5 \\3 & - 2 & 0\end{array} \right]$

Find a basis for the column space of the matrix. -Determine which of the following statements is false. A: The dimension of the vector space $\mathrm { P } _ { 7 }$ of polynomials is 8 . $B$ : Any line in $R ^ { 3 }$ is a one-dimensional subspace of $R ^ { 3 }$ . $C :$ If a vector space $\mathrm { V }$ has a basis $B = \left\{ b _ { 1 } , \ldots . . , b _ { 7 } \right\}$ , then any set in $\mathrm { V }$ containing 8 vectors must be linearly dependent.

Solve the problem. - $A = \left[ \begin{array} { r r r r r c r } 1 & - 2 & 3 & 1 & 0 & 5 & - 4 \\0 & 0 & 1 & - 6 & 2 & - 2 & 0 \\0 & 0 & 0 & 0 & 0 & 1 & 3 \\0 & 0 & 0 & 0 & 0 & 0 & 0\end{array} \right]$

Solve the problem. -Consider two bases $B = \left\{ \mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } \right\}$ and $C = \left\{ \mathbf { c } _ { 1 } , \mathbf { c } _ { 2 } \right\}$ for a vector space $V$ such that $\mathbf { b } _ { 1 } = \mathbf { c } _ { 1 } - 2 \mathbf { c } _ { 2 }$ and $\mathbf { b } _ { 2 } = 5 \mathbf { c } _ { 1 } + 4 \mathbf { c } _ { 2 }$ . Suppose $\mathbf { x } = \mathbf { b } _ { 1 } + 3 \mathbf { b } _ { 2 }$ . That is, suppose $[ \mathbf { x } ] _ { B } = \left[ \begin{array} { l } 1 \\ 3 \end{array} \right]$ . Find $[ \mathbf { x } ] _ { C }$ .

Find a basis for the set of all solutions to the difference equation. -Let $B = \left\{ \mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } \right\}$ and $C = \left\{ \mathbf { c } _ { 1 } , \mathbf { c } _ { 2 } \right\}$ be bases for $\mathcal { R } ^ { 2 }$ , where $\mathbf { b } _ { 1 } = \left[ \begin{array} { l } 2 \\4\end{array} \right] , \mathbf { b } _ { 2 } = \left[ \begin{array} { l } 1 \\3\end{array} \right] , \mathbf { c } _ { 1 } = \left[ \begin{array} { l } 1 \\3\end{array} \right] , \mathbf { c } _ { 2 } = \left[ \begin{array} { r } - 4 \\- 10\end{array} \right]$ Find the change-of-coordinates matrix from $B$ to $C$ .

Find the steady-state probability vector for the stochastic matrix P. - $P = \left[ \begin{array} { c c c } 0.9 & 0.3 & 0.4 \\0.1 & 0.6 & 0.1 \\0 & 0.1 & 0.5\end{array} \right]$

Determine whether the vector u belongs to the null space of the matrix A. - $\mathbf { u } = \left[ \begin{array} { l } 2 \\3 \\1\end{array} \right] , A = \left[ \begin{array} { r r r } - 2 & 3 & - 5 \\- 3 & - 1 & 9\end{array} \right]$

Determine whether {v₁, v₂, v₃} is a basis for R³ -Let $\mathrm { H } = \left\{ \left[ \begin{array} { c } \mathrm { a } + 3 \mathrm {~b} + 4 \mathrm {~d} \\ \mathrm { c } + \mathrm { d } \\ - 3 \mathrm { a } - 9 \mathrm {~b} + 4 \mathrm { c } - 8 \mathrm {~d} \\ - \mathrm { c } - \mathrm { d } \end{array} \right] : a , b , c , \mathrm {~d} \right.$ in $\left. \mathscr { R } \right\}$ Find the dimension of the subspace $\mathrm { H }$ .

Solve the problem. -Let $H$ be the set of all polynomials of the form $\mathrm { p } ( \mathrm { t } ) = \mathrm { a } + \mathrm { bt } ^ { 2 }$ where a and $\mathrm { b }$ are in $R$ and $b > a$ . Determine whether $\mathrm { H }$ is a vector space. If it is not a vector space, determine which of the following properties it fails to satisfy. A: Contains zero vector B: Closed under vector addition C: Closed under multiplication by scalars

Determine whether {v₁, v₂, v₃} is a basis for R³ - $B = \left\{ \left[ \begin{array} { r } - 2 \\2\end{array} \right] , \left[ \begin{array} { l } 0 \\1\end{array} \right] \right\} , [ \mathrm { x } ] _ { B } = \left[ \begin{array} { r } - 5 \\5\end{array} \right]$

Solve the problem. - $W$ is the set of all vectors of the form $\left[ \begin{array} { c } a - 4 b \\ 5 \\ 4 a + b \\ - a - b \end{array} \right]$ , where $a$ and $b$ are arbitrary real numbers.

Find the specified change-of-coordinates matrix. -Consider two bases $B = \left\{ \mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } \right\}$ and $C = \left\{ \mathbf { c } _ { 1 } , \mathbf { c } _ { 2 } \right\}$ for a vector space $V$ such that $\mathbf { b } _ { 1 } = \mathbf { c } _ { 1 } - 5 \mathbf { c } _ { 2 }$ and $\mathbf { b } _ { 2 } = 2 \mathbf { c } _ { 1 } - 4 c _ { 2 }$ . Find the change-of-coordinates matrix from $B$ to $C$ .

Determine if the vector u is in the column space of matrix A and whether it is in the null space of A. -Let $\mathbf { v } _ { \mathbf { 1 } } = \left[ \begin{array} { l } - 4 \\ - 1 \\ - 2 \end{array} \right] , \mathbf { v } _ { \mathbf { 2 } } = \left[ \begin{array} { r } - 3 \\ 1 \\ - 2 \end{array} \right] , \mathbf { v } _ { \mathbf { 3 } } = \left[ \begin{array} { r } 1 \\ - 5 \\ 2 \end{array} \right]$ , and $\mathrm { H } = \operatorname { Span } \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \right\}$ . Note that $\mathbf { v } _ { 3 } = 2 \mathbf { v } _ { 1 } - 3 \mathbf { v } _ { 2 }$ . Which of the following sets form a basis for the subspace $H$ , i.e., which sets form an efficient spanning set containing no unnecessary vectors? $A : \left\{ v _ { 1 } , v _ { 2 } , v _ { 3 } \right\}$ $B : \left\{ v _ { 1 } , v _ { 2 } \right\}$ $C : \left\{ v _ { 1 } , v _ { 3 } \right\}$ D: $\left\{ v _ { 2 } , v _ { 3 } \right\}$

Solve the problem. -Determine which of the following sets is a vector space. $V$ is the line $y = x$ in the $x y$ -plane: $V = \left\{ \left[ \begin{array} { l } x \\ y \end{array} \right] : y = x \right\}$ $W$ is the union of the first and second quadrants in the xy-plane: $W = \left\{ \left[ \begin{array} { l } x \\ y \end{array} \right] : y \geq 0 \right\}$ $U$ is the line $y = x + 1$ in the $x y$ -plane: $U = \left\{ \left[ \begin{array} { l } x \\ y \end{array} \right] : y = x + 1 \right\}$

Determine whether the signals are linearly independent. - $2 \mathrm { k } - 5,4 \mathrm { k } - 1 , \mathrm { k } + 3$

Solve the problem. -Determine which of the following sets is a subspace of $P _ { n }$ for an appropriate value of $n$ . A: All polynomials of the form $p ( t ) = a + b t ^ { 2 }$ , where $a$ and $b$ are in $R$ B: All polynomials of degree exactly 4 , with real coefficients C: All polynomials of degree at most 4 , with positive coefficients

Solve the problem. -A mathematician has found 5 solutions to a homogeneous system of 40 equations in 42 variables. The 5 solutions are linearly independent and all other solutions can be constructed by adding together appropriate multiples of these 5 solutions. Will the system necessarily have a solution for every possible choice of constants on the right side of the equation? Explain.

Solve the problem. - $A = \left[ \begin{array} { r r r r r } 1 & 3 & - 4 & 0 & 1 \\2 & 4 & - 5 & 2 & - 4 \\1 & - 5 & 0 & - 3 & 2 \\- 3 & - 1 & 8 & 3 & - 4\end{array} \right] , B = \left[ \begin{array} { r r r r r } 1 & 3 & - 4 & 0 & 1 \\0 & - 2 & 3 & 2 & - 6 \\0 & 0 & - 8 & - 11 & 25 \\0 & 0 & 0 & 0 & 0\end{array} \right]$