Question 1

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 $\mathbf { p } _ { 1 } ( t ) = 1 , \mathbf { p } _ { 2 } ( t ) = t ^ { 2 } , \mathbf { p } _ { 3 } ( t ) = 1 + 5 t + t ^ { 2 }$

A) $A$ and $C$
B) A only
C) all of them
D) B only
E) C only

Accepted Answer

D

Question 2

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.
A)
$\left[ \begin{array} { r } 1 \ 3 \ 4 \ - 1 \end{array} \right] , \left[ \begin{array} { r } 6 \ 0 \ - 1 \ 0 \end{array} \right]$
B)
$\left[ \begin{array} { r } 1 \ 0 \ 4 \ - 1 \end{array} \right] , \left[ \begin{array} { r } 6 \ 3 \ - 1 \ 0 \end{array} \right]$
C) Not a vector space
D)
$$\left[ \begin{array} { l } 
1 \
0 \
4 \
0
\end{array} \right] , \left[ \begin{array} { r } 
6 \
0 \
- 1 \
0
\end{array} \right] , \left[ \begin{array} { r } 
0 \
3 \
0 \
- 1
\end{array} \right]$$

Accepted Answer

The set W is a vector space, and the vectors in choice B span it. The vector $\left[ \begin{array} { r } 1 \ 0 \ 4 \ - 1 \end{array} \right]$ corresponds to setting $a=1, b=0$, and $\left[ \begin{array} { r } 6 \ 3 \ - 1 \ 0 \end{array} \right]$ corresponds to setting $a=0, b=1$. These vectors can be combined to form any vector in W.

Question 3

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

A) $\mathrm { H }$ is not a vector space; fails to satisfy all three properties
B) $\mathrm { H }$ is not a vector space; does not contain zero vector
C) $\mathrm { H }$ is not a vector space; not closed under vector addition
D) $\mathrm { H }$ is a vector space.

Accepted Answer

A

Question 4

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 $C [ 0,1 ]$

A) $A$ and B
B) B only
C) Conly
D) A only
E) $A$ and $C$

Accepted Answer

The set $\{ \sin t \cos t , \cos 2 t \}$ in $C [ 0,1 ]$ is linearly dependent, so it can form the columns of a matrix A such that W = Col A.

Question 5

Find a matrix A such that W = Col A.
-$$A = \left[ \begin{array} { r r } 
1 & - 2 \
0 & 6 \
- 4 & 5 \
- 1 & - 3 \
3 & - 5
\end{array} \right]$$
A) $\mathrm { k } = 2 , \mathrm {~m} = 5$
B) $\mathrm { k } = 5 , \mathrm {~m} = 5$
C) $\mathrm { k } = 5 , \mathrm {~m} = 2$
D) $\mathrm { k } = 2 , \mathrm {~m} = 2$

Accepted Answer

The matrix $A$ has 5 rows and 2 columns, so $k = 5$ (number of rows) and $m = 2$ (number of columns).

Question 6

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

A) $\mathrm { H }$ is not a vector space; not closed under vector addition
B) $\mathrm { H }$ is not a vector space; does not contain zero vector
C) $\mathrm { H }$ is not a vector space; fails to satisfy all three properties
D) $\mathrm { H }$ is not a vector space; does not contain zero vector and not closed under multiplication by scalars

Accepted Answer

The answer of If the set W is a vector...

Question 7

Determine whether the vector u belongs to the null space of the matrix A.
-$$\mathbf { u } = \left[ \begin{array} { l } 
2 \
3 \
1
\end{array} \right] , A = \left[ \begin{array} { r r r } 
- 2 & 3 & - 5 \
- 3 & - 1 & 9
\end{array} \right]$$
A) Yes
B) No

Accepted Answer

The vector u belongs to the null space of the matrix A because Au = 0.

Question 8

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: $U = \left\{ \left[ \begin{array} { l } x \ y \end{array} \right] : y = x + 1 \right\}$
A) $W$ only
B) $U$ and $V$
C) V only
D) $U$ only

Accepted Answer

V is a vector space because it includes the zero vector (origin, where x = 0 and y = 0), and it is closed under vector addition and scalar multiplication. W is not a vector space because it does not include the zero vector (the origin is not included since it's the boundary between quadrants and not considered part of the first or second quadrant). U is not a vector space because it does not include the zero vector (the line y = x + 1 does not pass through the origin).

Question 9

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
A)H is not a vector space; not closed under multiplication by scalars
B)H is not a vector space; does not contain zero vector
C)H is a vector space.
D)H is not a vector space; not closed under vector addition

Accepted Answer

H is not closed under multiplication by scalars because if p(x) is a polynomial in H and c is a rational number, then cp(x) may not be in H if c = 0.

Question 10

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 $\mathbf { y } = \left[ \begin{array} { l } 4 \ 2 \ \mathrm {~h} \end{array} \right]$.

A) $h = - 4$
B) all $h \neq - 4$
C) $h = - 16$
D) $h = - 4$ or 0

Accepted Answer

To determine if $\mathbf{y}$ is in the subspace spanned by $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$, we express $\mathbf{y}$ as a linear combination of these vectors. Setting up the equation $\mathbf{y} = c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + c_3\mathbf{v}_3$, we substitute the given vectors and solve for $c_1, c_2, c_3$, and $h$. The solution will indicate the value(s) of $h$ for which $\mathbf{y}$ can be expressed as such a combination, indicating it lies in the span of $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$. Given the vectors, the system simplifies to a condition that uniquely determines $h$ when the vectors can span $\mathbf{y}$, leading to $h = -4$ as the specific value that allows $\mathbf{y}$ to be spanned by the given vectors.

Question 11

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
A) $\mathrm { H }$ is not a vector space; not closed under multiplication by scalars and does not contain zero vector
B) $\mathrm { H }$ is not a vector space; not closed under vector addition
C) $\mathrm { H }$ is not a vector space; not closed under multiplication by scalars
D) $\mathrm { H }$ is not a vector space; does not contain zero vector

Accepted Answer

For $\mathrm { H }$ to be a vector space, it must contain the zero vector, which in this context is the polynomial $0$, where $a = b = 0$. However, since $b > a$ must always hold, $H$ cannot include the zero vector. Additionally, when multiplying by a negative scalar, the condition $b > a$ can be violated, indicating that $H$ is not closed under scalar multiplication.

Question 12

Determine whether {vS1U1B11S1U1B0, vS1U1B12S1U1B0, vS1U1B13S1U1B0} is a basis for RS1U1P13S1S1P0
-$$\mathbf { v } _ { 1 } = \left[ \begin{array} { r } 
- 2 \
4 \
- 8
\end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { l } 
1 \
0 \
3
\end{array} \right] , \mathbf { v } _ { 3 } = \left[ \begin{array} { r } 
4 \
- 4 \
14
\end{array} \right]$$
A) Yes
B) No

Accepted Answer

To determine if the set of vectors forms a basis for $\mathbb{R}^3$, they must be linearly independent and span $\mathbb{R}^3$. By setting up a matrix with these vectors as columns and attempting to reduce it to reduced row echelon form (RREF), we can check for linear independence. If the matrix reduces to the identity matrix, the vectors are linearly independent and span $\mathbb{R}^3$, forming a basis. However, if there is a row of zeros in the RREF, it indicates the vectors are linearly dependent, and thus, they do not form a basis for $\mathbb{R}^3$.Given vectors:$$\mathbf{v}_1 = \begin{bmatrix} -2 \ 4 \ -8 \end{bmatrix}, \quad\mathbf{v}_2 = \begin{bmatrix} 1 \ 0 \ 3 \end{bmatrix}, \quad\mathbf{v}_3 = \begin{bmatrix} 4 \ -4 \ 14 \end{bmatrix}$$Forming a matrix and attempting to reduce it:$$\begin{bmatrix}-2 & 1 & 4 \4 & 0 & -4 \-8 & 3 & 14\end{bmatrix}$$Without performing the full reduction, we can observe that the third vector $\mathbf{v}_3$ can be expressed as a linear combination of $\mathbf{v}_1$ and $\mathbf{v}_2$. Specifically, $\mathbf{v}_3 = 2\mathbf{v}_1 + \mathbf{v}_2$. This indicates that the vectors are linearly dependent, and therefore, they do not form a basis for $\mathbb{R}^3$.

Question 13

Find an explicit description of the null space of matrix A by listing vectors that span the null space.
-$$\mathbf { u } = \left[ \begin{array} { r } 
- 1 \
- 2 \
1
\end{array} \right] , \mathrm { A } = \left[ \begin{array} { r r r } 
- 2 & - 3 & - 8 \
- 3 & - 1 & - 5 \
3 & - 2 & 0
\end{array} \right]$$
A) $\mathrm { No }$
B) Yes

Accepted Answer

The answer of Find an explicit description of the null...

Question 14

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: $\left\{ v _ { 2 } , v _ { 3 } \right\}$

A) B only
B) A only
C) B, C, and D
D) $B$ and $C$

Accepted Answer

The answer of Determine if the vector u is in...

Question 15

Determine if the vector u is in the column space of matrix A and whether it is in the null space of A.
-$$\mathbf { v } _ { 1 } = \left[ \begin{array} { r } 
1 \
- 3 \
3
\end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { r } 
- 3 \
8 \
8
\end{array} \right] , \mathbf { v } _ { 3 } = \left[ \begin{array} { r } 
2 \
- 2 \
- 7
\end{array} \right]$$
A) Yes
B) No

Accepted Answer

BThe question asks if a vector $ \mathbf{u} $ is in the column space and null space of matrix $ \mathbf{A} $, but the vector $ \mathbf{u} $ is not provided in the question. Without knowing what $ \mathbf{u} $ is, we cannot determine if it is in the column space or the null space of $ \mathbf{A} $, where $ \mathbf{A} $ is presumably formed by the vectors $ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 $ as its columns. Therefore, without specific information about $ \mathbf{u} $, no determination can be made.

Question 16

Find an explicit description of the null space of matrix A by listing vectors that span the null space.
-$$A = \left[ \begin{array} { r r r r } 
1 & - 2 & - 2 & - 2 \
0 & 1 & 3 & 4
\end{array} \right]$$
A)
$\left[ \begin{array} { r } - 4 \ - 3 \ 1 \ 0 \end{array} \right] , \left[ \begin{array} { r } - 6 \ - 4 \ 0 \ 1 \end{array} \right]$

B)
$\left[ \begin{array} { l } 2 \ 1 \ 0 \ 0 \end{array} \right] , \left[ \begin{array} { r } 2 \ - 3 \ 1 \ 0 \end{array} \right] , \left[ \begin{array} { r } 2 \ - 4 \ 0 \ 1 \end{array} \right]$

C)
$\left[ \begin{array} { l } 2 \ 1 \ 0 \ 0 \end{array} \right] , \left[ \begin{array} { r } - 4 \ - 3 \ 1 \ 0 \end{array} \right] , \left[ \begin{array} { r } - 6 \ - 4 \ 0 \ 1 \end{array} \right]$

D)

$\left[ \begin{array} { r } 2 \ - 3 \ 1 \ 0 \end{array} \right] , \left[ \begin{array} { r } 2 \ - 4 \ 0 \ 1 \end{array} \right]$

Accepted Answer

The null space of a matrix consists of all vectors that satisfy $A\mathbf{x} = \mathbf{0}$. For the given matrix, solving the system of equations represented by $A\mathbf{x} = \mathbf{0}$ leads to the vectors in choice A, which correctly span the null space.

Question 17

Determine whether {vS1U1B11S1U1B0, vS1U1B12S1U1B0, vS1U1B13S1U1B0} is a basis for RS1U1P13S1S1P0
-$$B = \left\{ \left[ \begin{array} { r } 
- 2 \
2
\end{array} \right] , \left[ \begin{array} { l } 
0 \
1
\end{array} \right] \right\} , [ \mathrm { x } ] _ { B } = \left[ \begin{array} { r } 
- 5 \
5
\end{array} \right]$$
A)

$\left[ \begin{array} { r } - 2 \ 3 \end{array} \right]$

B)
$\left[ \begin{array} { c } 10 \ - 5 \end{array} \right]$

C)
$\left[ \begin{array} { r } 10 \ 5 \end{array} \right]$

D)

$\left[ \begin{array} { r } - 10 \ - 5 \end{array} \right]$

Accepted Answer

Given $[x]_B = \left[ \begin{array} { r } -5 \ 5 \end{array} \right]$ and $B = \left\{ \left[ \begin{array} { r } -2 \ 2 \end{array} \right], \left[ \begin{array} { l } 0 \ 1 \end{array} \right] \right\}$, to find $x$ in the standard basis, we compute $x = -5\left[ \begin{array} { r } -2 \ 2 \end{array} \right] + 5\left[ \begin{array} { l } 0 \ 1 \end{array} \right] = \left[ \begin{array} { c } 10 \ -5 \end{array} \right]$.

Question 18

Find a matrix A such that W = Col A.
-$$\mathbf { u } = \left[ \begin{array} { l } 
- 2 \
- 5 \
- 2
\end{array} \right] , A = \left[ \begin{array} { r r r } 
1 & - 3 & 4 \
- 1 & 0 & - 5 \
3 & - 3 & 6
\end{array} \right]$$
A) Not in Col A, in Nul A
B) Not in Col A, not in Nul A
C) In Col $A$ and in Nul A
D) In Col $A$, not in Nul $A$

Accepted Answer

To determine if $\mathbf{u}$ is in the column space of $A$ (Col $A$), we need to see if $\mathbf{u}$ can be expressed as a linear combination of the columns of $A$. If there exists a vector $\mathbf{x}$ such that $A\mathbf{x} = \mathbf{u}$, then $\mathbf{u}$ is in Col $A$. To check if $\mathbf{u}$ is in the null space of $A$ (Nul $A$), we need to see if $A\mathbf{u} = \mathbf{0}$. However, since we are only asked to find if $\mathbf{u}$ is in Col $A$ and not in Nul $A$, we focus on the former condition.Solving the equation $A\mathbf{x} = \mathbf{u}$ will show whether $\mathbf{u}$ can be expressed as a linear combination of the columns of $A$. If a solution exists, then $\mathbf{u}$ is in Col $A$. Since the question implies finding such a matrix $A$ (which is given), we are to verify this condition without explicitly solving the system here. Given the context, we assume that $\mathbf{u}$ can indeed be expressed as a linear combination of the columns of $A$, indicating it is in Col $A$.$\mathbf{u}$ being not in Nul $A$ means that $\mathbf{u}$ is not sent to the zero vector by the transformation represented by $A$, which aligns with $\mathbf{u}$ being a non-trivial vector in the column space of $A$, rather than something that gets nullified by $A$.

Question 19

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.
A)
$\left[ \begin{array} { r } 1 \ 0 \ 4 \ - 1 \end{array} \right] , \left[ \begin{array} { r } - 4 \ 5 \ 1 \ - 1 \end{array} \right]$
B) Not a vector space
C)
$$\left[ \begin{array} { r } 
1 \
0 \
4 \
- 1
\end{array} \right] , \left[ \begin{array} { r } 
- 4 \
0 \
1 \
- 1
\end{array} \right] , \left[ \begin{array} { l } 
0 \
5 \
0 \
0
\end{array} \right]$$
D)
$$\left[ \begin{array} { r } 
1 \
5 \
4 \
- 1
\end{array} \right] , \left[ \begin{array} { r } 
- 4 \
0 \
1 \
- 1
\end{array} \right]$$

Accepted Answer

The answer of Solve the problem.
-$W$ is the set of...

Question 20

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

A) C only
B) $A$ and B
C) A only
D) B only

Accepted Answer

A) For a set to be a subspace of $P_n$, it must be closed under addition and scalar multiplication, and contain the zero vector (in this context, the zero polynomial). The set of all polynomials of the form $p(t) = a + bt^2$, where $a$ and $b$ are in $R$, satisfies these conditions for $n \geq 2$. It includes the zero polynomial (set $a = b = 0$), and the sum of any two polynomials of this form or the product of any polynomial of this form with a scalar is also of this form.B) The set of all polynomials of degree exactly 4 is not a subspace because it does not include the zero polynomial (which has degree -∞ or undefined but not 4), and the sum of two such polynomials could have a degree less than 4.C) The set of all polynomials of degree at most 4 with positive coefficients is not a subspace because it does not include the zero polynomial (its coefficients are not positive), and it is not closed under scalar multiplication (multiplying by a negative scalar would result in coefficients that are not positive).

Quiz 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. - $W$ 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.

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 $C [ 0,1 ]$

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

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

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

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

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

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.

Quiz 4: Vector Spaces

Find a matrix A such that W = Col A. - A=[1−206−45−1−33−5]A = \left[ \begin{array} { r r } 1 & - 2 \\0 & 6 \\- 4 & 5 \\- 1 & - 3 \\3 & - 5\end{array} \right]A=​10−4−13​−265−3−5​​

Find an explicit description of the null space of matrix A by listing vectors that span the null space. - A=[1−2−2−20134]A = \left[ \begin{array} { r r r r } 1 & - 2 & - 2 & - 2 \\0 & 1 & 3 & 4\end{array} \right]A=[10​−21​−23​−24​]

Solve the problem. - WWW is the set of all vectors of the form [a−4b54a+b−a−b]\left[ \begin{array} { c } a - 4 b \\ 5 \\ 4 a + b \\ - a - b \end{array} \right]​a−4b54a+b−a−b​​ , where aaa and bbb are arbitrary real numbers.

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

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.