Exam 4: Vector Spaces

If the set W is a vector space, find a set S of vectors that spans it. Otherwise, state that W is not a vector space. -Let $\mathrm { H }$ be the set of all points in the $\mathrm { xy }$ -plane having at least one nonzero coordinate: $\mathrm { H } = \left\{ \left[ \begin{array} { l } \mathrm { x } \\ \mathrm { y } \end{array} \right] : \mathrm { x } , \mathrm { y } \right.$ not both zero $\}$ . 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

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Question 1

Correct Answer:

Verified

C

Solve the problem. -Let $\mathrm { H }$ be the set of all points of the form $( \mathrm { s } , \mathrm { s } - 1 )$ . 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

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Question 2

Correct Answer:

Verified

A

Solve the problem. -Suppose a nonhomogeneous system of 15 linear equations in 17 unknowns has a solution for all possible constants on the right side of the equation. Is it possible to find 3 nonzero solutions of the associated homogeneous system that are linearly independent? Explain.

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Question 3

Correct Answer:

Verified

If the null space of a $5 \times 7$ matrix is 3 -dimensional, find Rank A, Dim Row A, and Dim Col A.
A) Rank A = 4, Dim Row A = 4, Dim Col A $= 3$
B) Rank $\mathrm { A } = 4 , \operatorname { Dim }$ Row $\mathrm { A } = 4$ , $\operatorname { Dim } \mathrm { Col } \mathrm {} \mathrm { A } = 4$
C) Rank A $= 4 , \operatorname { Dim } \operatorname { Row } \mathrm { A } = 3$ , $\operatorname { Dim } \operatorname { Col~A } = 3$
D) $\operatorname { Rank } A = 2 , \operatorname { Dim } \operatorname { Row } A = 2$ , $\operatorname { Dim } \operatorname { Col~A } = 2$

Determine whether {v₁, v₂, v₃} is a basis for R³ -Let $\mathrm { A } = \left[ \begin{array} { r r r r r } - 1 & 3 & 7 & 2 & 0 \\ 1 & - 2 & - 7 & - 1 & 3 \\ 2 & - 4 & - 9 & - 5 & 1 \\ 3 & - 6 & - 11 & - 9 & - 1 \end{array} \right]$ and $B = \left[ \begin{array} { r r r r r } 1 & - 3 & - 7 & - 2 & 0 \\ 0 & 1 & 0 & 1 & 3 \\ 0 & 0 & 5 & - 3 & - 5 \\ 0 & 0 & 0 & 0 & 0 \end{array} \right]$ It can be shown that matrix $\mathrm { A }$ is row equivalent to matrix $\mathrm { B }$ . Find a basis for $\mathrm { Col } \mathrm {} \mathrm { A }$ .

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Question 4

Find the new coordinate vector for the vector x after performing the specified change of basis. -If $\mathrm { A }$ is a $5 \times 9$ matrix, what is the smallest possible dimension of Nul $\mathrm { A }$ ?

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Question 5

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

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Question 6

Find the general solution of the difference equation. -Given that the signal $\mathrm { y } _ { \mathrm { k } } = \mathrm { k } ^ { 2 }$ is a solution of the given difference equation, find a description of all solutions of the equation. $\mathrm { y } \mathrm { k } + 2 + 6 \mathrm { y } \mathrm { k } + 1 - 7 \mathrm { yk } = 16 \mathrm { k } + 10$ for all $\mathrm { k }$

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Question 7

Determine whether the vector u belongs to the null space of the matrix A. -Find all values of $h$ such that $\mathbf { y }$ will be in the subspace of $\mathfrak { R } ^ { 3 }$ spanned by $\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 }$ if $\mathbf { v } _ { 1 } = \left[ \begin{array} { r } 1 \\ 2 \\ - 4 \end{array} \right]$ , $\mathbf { v } _ { 2 } = \left[ \begin{array} { r } 3 \\ 4 \\ - 8 \end{array} \right] , \mathbf { v } _ { 3 } = \left[ \begin{array} { r } - 1 \\ 0 \\ 0 \end{array} \right]$ , and $\mathbf { y } = \left[ \begin{array} { l } 4 \\ 2 \\ \mathrm {~h} \end{array} \right]$ .

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Question 8

Find a matrix A such that W = Col A. -A: The set $\left\{ \mathrm { p } _ { 1 } , \mathrm { p } _ { 2 } , \mathrm { p } _ { 3 } \right\}$ where $\mathrm { p } _ { 1 } ( \mathrm { t } ) = 1 , \mathrm { p } _ { 2 } ( \mathrm { t } ) = \mathrm { t } ^ { 2 } , \mathrm { p } _ { 3 } ( \mathrm { t } ) = 1 + 5 \mathrm { t }$ B: The set $\left\{ p _ { 1 } , p _ { 2 } , p _ { 3 } \right\}$ where $p _ { 1 } ( t ) = t , p _ { 2 } ( t ) = t ^ { 2 } , p _ { 3 } ( t ) = 2 t + 5 t ^ { 2 }$ $C :$ The set $\left\{ \mathbf { p } _ { 1 } , \mathbf { p } _ { 2 } , \mathbf { p } _ { 3 } \right\}$ where $\mathbf { p } _ { 1 } ( t ) = 1 , \mathbf { p } _ { 2 } ( t ) = t ^ { 2 } , \mathbf { p } _ { 3 } ( t ) = 1 + 5 t + t ^ { 2 }$

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Question 9

Find a matrix A such that W = Col A. -A: The set $\{ \sin t , \tan t \}$ in $C [ 0,1 ]$ B: The set $\{ \sin t \cos t , \cos 2 t \}$ in $C [ 0,1 ]$ $C :$ The set $\left\{ \cos ^ { 2 } t , 1 + \cos 2 t \right\}$ in $C [ 0,1 ]$

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Question 10

Determine whether {v₁, v₂, v₃} is a basis for R³ -Given the set of vectors $\left\{ \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right] , \left[ \begin{array} { l } 0 \\ 1 \\ 2 \end{array} \right] \right\}$ , decide which of the following statements is true: A: Set is linearly independent and spans $R ^ { 3 }$ . Set is a basis for $R ^ { 3 }$ . B: Set is linearly independent but does not span $R ^ { 3 }$ . Set is not a basis for $R ^ { 3 }$ . C: Set spans $\mathscr { R } ^ { 3 }$ but is not linearly independent. Set is not a basis for $R ^ { 3 }$ . D: Set is not linearly independent and does not span $R ^ { 3 }$ . Set is not a basis for $R ^ { 3 }$ .

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Question 11

Determine whether the signals are linearly independent. - $1 \mathrm { k } , 2 \mathrm { k } , ( - 3 ) ^ { \mathrm { k } }$

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Question 12

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

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Question 13

If the set W is a vector space, find a set S of vectors that spans it. Otherwise, state that W is not a vector space. - $W$ is the set of all vectors of the form $\left[ \begin{array} { c } a + 6 b \\ 3 b \\ 4 a - b \\ - a \end{array} \right]$ , where $a$ and $b$ are arbitrary real numbers.

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Question 14

Solve the problem. -Let H be the set of all polynomials having degree at most 4 and rational coefficients. Determine whether 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

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Question 15

Assume that the matrix A is row equivalent to B. Find a basis for the row space of the matrix A. - $A = \left[ \begin{array} { r r r r r } 1 & 5 & 1 & 3 & 0 \\- 2 & - 4 & - 5 & - 4 & 1\end{array} \right]$

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Question 16

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 $B = \left\{ \mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } \right\}$ $Determine whether {v1, v2, v3} is a basis for \mathbf { b } _ { 1 } = \left[ \begin{array} { l } 1 \\ 1 \end{array} \right] , \mathbf { b } _ { 2 } = \left[ \begin{array} { r } 1 \\ - 1 \end{array} \right] , \mathbf { x } = \left[ \begin{array} { r } 3 \\ - 5 \end{array} \right] , and B = \left\{ \mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } \right\} - B = \left\{ \left[ \begin{array} { r } 1 \\ - 3 \\ - 3 \end{array} \right] , \left[ \begin{array} { r } - 3 \\ 8 \\ - 3 \end{array} \right] , \left[ \begin{array} { r } 2 \\ - 2 \\ 2 \end{array} \right] \right\} , [ \mathrm { x } ] B = \left[ \begin{array} { r } - 4 \\ 2 \\ - 3 \end{array} \right]$ - $B = \left\{ \left[ \begin{array} { r } 1 \\- 3 \\- 3\end{array} \right] , \left[ \begin{array} { r } - 3 \\8 \\- 3\end{array} \right] , \left[ \begin{array} { r } 2 \\- 2 \\2\end{array} \right] \right\} , [ \mathrm { x } ] B = \left[ \begin{array} { r } - 4 \\2 \\- 3\end{array} \right]$

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Question 17

Find the specified change-of-coordinates matrix. - $\mathrm { y } _ { \mathrm { k } + 2 } + 5 \mathrm { y } _ { \mathrm { k } + 1 } - 6 \mathrm { y } _ { \mathrm { k } } = 0$ for all $\mathrm { k }$

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Question 18

Solve the problem. -Suppose that demographic studies show that each year about 6% of a cityʹs population moves to the suburbs (and 94% 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% of the population of the region lived in the City and 36% lived in the suburbs. What is the distribution of the population in 2002? For Simplicity, ignore other influences on the population such as births, deaths, and migration into and Out of the city/suburban region.

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Question 19

Determine whether {v₁, v₂, v₃} is a basis for R³ - $\mathbf { v } _ { 1 } = \left[ \begin{array} { r } - 2 \\4 \\- 8\end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { l } 1 \\0 \\3\end{array} \right] , \mathbf { v } _ { 3 } = \left[ \begin{array} { r } 4 \\- 4 \\14\end{array} \right]$

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Question 20

Solve the problem. -Suppose a nonhomogeneous system of 15 linear equations in 17 unknowns has a solution for all possible constants on the right side of the equation. Is it possible to find 3 nonzero solutions of the associated homogeneous system that are linearly independent? Explain.

Find the new coordinate vector for the vector x after performing the specified change of basis. -If $\mathrm { A }$ is a $5 \times 9$ matrix, what is the smallest possible dimension of Nul $\mathrm { A }$ ?

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

Find a matrix A such that W = Col A. -A: The set $\{ \sin t , \tan t \}$ in $C [ 0,1 ]$ B: The set $\{ \sin t \cos t , \cos 2 t \}$ in $C [ 0,1 ]$ $C :$ The set $\left\{ \cos ^ { 2 } t , 1 + \cos 2 t \right\}$ in $C [ 0,1 ]$

Determine whether the signals are linearly independent. - $1 \mathrm { k } , 2 \mathrm { k } , ( - 3 ) ^ { \mathrm { k } }$

If the set W is a vector space, find a set S of vectors that spans it. Otherwise, state that W is not a vector space. - $W$ is the set of all vectors of the form $\left[ \begin{array} { c } a + 6 b \\ 3 b \\ 4 a - b \\ - a \end{array} \right]$ , where $a$ and $b$ are arbitrary real numbers.

Assume that the matrix A is row equivalent to B. Find a basis for the row space of the matrix A. - $A = \left[ \begin{array} { r r r r r } 1 & 5 & 1 & 3 & 0 \\- 2 & - 4 & - 5 & - 4 & 1\end{array} \right]$

Find the specified change-of-coordinates matrix. - $\mathrm { y } _ { \mathrm { k } + 2 } + 5 \mathrm { y } _ { \mathrm { k } + 1 } - 6 \mathrm { y } _ { \mathrm { k } } = 0$ for all $\mathrm { k }$

Linear Equations in Linear Algebra

Matrix Algebra

Determinants

Eigenvalues and Eigenvectors

Orthogonality and Least Squares

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Exam 4: Vector Spaces

Solve the problem. -Suppose a nonhomogeneous system of 15 linear equations in 17 unknowns has a solution for all possible constants on the right side of the equation. Is it possible to find 3 nonzero solutions of the associated homogeneous system that are linearly independent? Explain.

Find the new coordinate vector for the vector x after performing the specified change of basis. -If A\mathrm { A }A is a 5×95 \times 95×9 matrix, what is the smallest possible dimension of Nul A\mathrm { A }A ?

Determine whether the signals are linearly independent. - 1k,2k,(−3)k1 \mathrm { k } , 2 \mathrm { k } , ( - 3 ) ^ { \mathrm { k } }1k,2k,(−3)k

Assume that the matrix A is row equivalent to B. Find a basis for the row space of the matrix A. - A=[15130−2−4−5−41]A = \left[ \begin{array} { r r r r r } 1 & 5 & 1 & 3 & 0 \\- 2 & - 4 & - 5 & - 4 & 1\end{array} \right]A=[1−2​5−4​1−5​3−4​01​]

Find the specified change-of-coordinates matrix. - yk+2+5yk+1−6yk=0\mathrm { y } _ { \mathrm { k } + 2 } + 5 \mathrm { y } _ { \mathrm { k } + 1 } - 6 \mathrm { y } _ { \mathrm { k } } = 0yk+2​+5yk+1​−6yk​=0 for all k\mathrm { k }k

Linear Equations in Linear Algebra

Matrix Algebra

Determinants

Eigenvalues and Eigenvectors

Orthogonality and Least Squares

Symmetric Matrices and Quadratic Forms

The Geometry of Vector Spaces

Optimization Online Only

Filters

Find the new coordinate vector for the vector x after performing the specified change of basis. -If $\mathrm { A }$ is a $5 \times 9$ matrix, what is the smallest possible dimension of Nul $\mathrm { A }$ ?

Determine whether the signals are linearly independent. - $1 \mathrm { k } , 2 \mathrm { k } , ( - 3 ) ^ { \mathrm { k } }$

Assume that the matrix A is row equivalent to B. Find a basis for the row space of the matrix A. - $A = \left[ \begin{array} { r r r r r } 1 & 5 & 1 & 3 & 0 \\- 2 & - 4 & - 5 & - 4 & 1\end{array} \right]$

Find the specified change-of-coordinates matrix. - $\mathrm { y } _ { \mathrm { k } + 2 } + 5 \mathrm { y } _ { \mathrm { k } + 1 } - 6 \mathrm { y } _ { \mathrm { k } } = 0$ for all $\mathrm { k }$