Question 1

Find the indicated vector.
-Let $\mathbf { u } = \left[ \begin{array} { l } 6 \ 1 \end{array} \right]$. Find $5 \mathbf { u }$.

Accepted Answer

A) 
$\left[ \begin{array} { r } - 30 \ - 5 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r } 30 \ 5 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r } - 30 \ 5 \end{array} \right]$ 
D) 
$\left[ \begin{array} { c } 30 \ - 5 \end{array} \right]$ 
A) 
$\left[ \begin{array} { r } - 30 \ - 5 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r } 30 \ 5 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r } - 30 \ 5 \end{array} \right]$ 
D) 
$\left[ \begin{array} { c } 30 \ - 5 \end{array} \right]$

Question 2

Compute the product or state that it is undefined.
-$$\left[ \begin{array} { r r r } 
- 2 & - 2 & 6 \
5 & 8 & - 5
\end{array} \right] \left[ \begin{array} { r } 
8 \
- 3 \
3
\end{array} \right]$$

Accepted Answer

A) 
$\left[ \begin{array} { l } 8 \ 1 \end{array} \right]$ 
B) 
$\left[ \begin{array} { l } 1 \ 8 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r } - 2 & - 2 & 6 \ 5 & 8 & - 5 \ 8 & - 3 & 3 \end{array} \right]$ 
D)

[8r 1 1] 
A) 
$\left[ \begin{array} { l } 8 \ 1 \end{array} \right]$ 
B) 
$\left[ \begin{array} { l } 1 \ 8 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r } - 2 & - 2 & 6 \ 5 & 8 & - 5 \ 8 & - 3 & 3 \end{array} \right]$ 
D)

[8r 1 1]

Question 3

Display the indicated vector(s) on an xy-graph.
-$$\text { Let } \mathbf { u } = \left[ \begin{array} { r } 
5 \
- 4
\end{array} \right] \text { Display the vector } 2 \mathbf { u } \text { using the given axes. }$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 4

Find the standard matrix of the linear transformation T.
-T: $\mathfrak { R } ^ { 2 } \rightarrow > \mathfrak { R } ^ { 2 }$ rotates points (about the origin) through $\frac { 7 } { 4 } \pi$ radians (with counterclockwise rotation for a positive angle).

Accepted Answer

A) 
$\left[\begin{array}{c}
\frac{\sqrt{3}}{3} \frac{\sqrt{3}}{3} \
-\frac{\sqrt{3}}{3} \frac{\sqrt{3}}{3}
\end{array}\right]$ 
B)  $\left[\begin{array}{c}
-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2} \
-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}
\end{array}\right]$ 
C) 
  
D)  
$\left[ \begin{array} { r r } 1 & 1 \ - 1 & 1 \end{array} \right]$ 
A) 
$\left[\begin{array}{c}
\frac{\sqrt{3}}{3} \frac{\sqrt{3}}{3} \
-\frac{\sqrt{3}}{3} \frac{\sqrt{3}}{3}
\end{array}\right]$ 
B)  $\left[\begin{array}{c}
-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2} \
-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}
\end{array}\right]$ 
C) 
  
D)  
$\left[ \begin{array} { r r } 1 & 1 \ - 1 & 1 \end{array} \right]$

Question 5

Solve the system of equations.
-$$\begin{array} { r } 
2 x _ { 1 } + x _ { 2 } = 0 \
x _ { 1 } - 3 x _ { 2 } + x _ { 3 } = 0 \
3 x _ { 1 } + x _ { 2 } - x _ { 3 } = 0
\end{array}$$

Accepted Answer

A)  $( 1,0,0 )$ 
B)  $( 0,1,0 )$ 
C)  No solution 
D)  $( 0,0,0 )$ 
A)  $( 1,0,0 )$ 
B)  $( 0,1,0 )$ 
C)  No solution 
D)  $( 0,0,0 )$

Question 6

Describe geometrically the effect of the transformation T.
-Let $\mathrm { A } = \left[ \begin{array} { l l } 1 & 0 \ 2 & 1 \end{array} \right]$.
Define a transformation $\mathrm { T }$ by $\mathrm { T } ( \mathrm { x } ) = \mathrm { Ax }$.

Accepted Answer

A)  Vertical shear 
B)  Projection onto $x _ { 2 }$-axis 
C)  Projection onto $\mathrm { x } _ { 1 }$-axis 
D)  Horizontal shear 
A)  Vertical shear 
B)  Projection onto $x _ { 2 }$-axis 
C)  Projection onto $\mathrm { x } _ { 1 }$-axis 
D)  Horizontal shear

Question 7

Solve the system of equations.
-$$\begin{aligned}
x _ { 1 } - x _ { 2 } + 8 x _ { 3 } & = - 107 \
6 x _ { 1 } + x _ { 3 } & = 17 \
3 x _ { 2 } - 5 x _ { 3 } & = 89
\end{aligned}$$

Accepted Answer

A)  $( 5,8 , - 13 )$ 
B)  $( 5 , - 8 , - 13 )$ 
C)  $( - 5,8,13 )$ 
D)  $( - 5 , - 8,13 )$ 
A)  $( 5,8 , - 13 )$ 
B)  $( 5 , - 8 , - 13 )$ 
C)  $( - 5,8,13 )$ 
D)  $( - 5 , - 8,13 )$

Question 8

Determine whether the system is consistent.
-$$\begin{aligned}
5 x _ { 2 } + x _ { 4 } & = - 11 \
x _ { 1 } + x _ { 2 } + 6 x _ { 3 } - x _ { 4 } & = 15 \
5 x _ { 1 } + x _ { 3 } + 6 x _ { 4 } & = 16 \
x _ { 1 } + x _ { 2 } + 3 x _ { 3 } & = 8
\end{aligned}$$

Accepted Answer

A)  $\mathrm { Yes }$ 
B)  $\mathrm { No }$ 
A)  $\mathrm { Yes }$ 
B)  $\mathrm { No }$

Question 9

Solve the system of equations.
-$$\begin{array} { l } 
5 x _ { 1 } + 2 x _ { 2 } + x _ { 3 } = - 11 \
2 x _ { 1 } - 3 x _ { 2 } - x _ { 3 } = 17 \
7 x _ { 1 } + x _ { 2 } + 2 x _ { 3 } = - 4
\end{array}$$

Accepted Answer

A)  $( 0 , - 6,1 )$ 
B)  $( - 3,0,4 )$ 
C)  $( 0,6 , - 1 )$ 
D)  $( 3,0 , - 4 )$ 
A)  $( 0 , - 6,1 )$ 
B)  $( - 3,0,4 )$ 
C)  $( 0,6 , - 1 )$ 
D)  $( 3,0 , - 4 )$

Question 10

Compute the product or state that it is undefined.
-$$\left[ \begin{array} { r r } 
5 & - 3 \
- 3 & 4 \
- 3 & 8
\end{array} \right] \left[ \begin{array} { r } 
2 \
- 4 \
8
\end{array} \right]$$

Accepted Answer

A)  Undefined 
B) 
$\left[\begin{array}{r}
4 \
-4 \
40
\end{array}\right]$ 
C) 
$\left[\begin{array}{ll}
-2 & 42
\end{array}\right]$ 
D) 
$\left[\begin{array}{rr}
10 & -6 \
12 & -16 \
-24 & 64
\end{array}\right]$ 
A)  Undefined 
B) 
$\left[\begin{array}{r}
4 \
-4 \
40
\end{array}\right]$ 
C) 
$\left[\begin{array}{ll}
-2 & 42
\end{array}\right]$ 
D) 
$\left[\begin{array}{rr}
10 & -6 \
12 & -16 \
-24 & 64
\end{array}\right]$

Question 11

Determine whether the system is consistent.
-$$\begin{array} { l } 
5 x _ { 1 } + 2 x _ { 2 } + x _ { 3 } = - 11 \
2 x _ { 1 } - 3 x _ { 2 } - x _ { 3 } = 17 \
7 x _ { 1 } + x _ { 2 } + 2 x _ { 3 } = - 4
\end{array}$$

Accepted Answer

A)  No 
B)  Yes 
A)  No 
B)  Yes

Question 12

Solve the problem.
-A company manufactures two products. For $1.00 worth of product A, the company 
spends $0.50 on materials, $0.20 on labor, and $0.15 on overhead. For $1.00 worth of
product B, the company spends $0.45 on materials, $0.20 on labor, and $0.15 on overhead.
Let $$\mathbf { a } = \left[ \begin{array} { l } 
0.50 \
0.20 \
0.15
\end{array} \right] \text { and } \mathbf { b } = \left[ \begin{array} { l } 
0.45 \
0.20 \
0.15
\end{array} \right]$$ $$\begin{array} { l } 
x _ { 1 } a + x _ { 2 } \mathbf { b } = \left[ \begin{array} { r } 
140 \
60 \
45
\end{array} \right] \
\text { or } \
x _ { 1 } \left[ \begin{array} { l } 
0.50 \
0.20 \
0.15
\end{array} \right] + x _ { 2 } \left[ \begin{array} { l } 
0.45 \
0.20 \
0.15
\end{array} \right] = \left[ \begin{array} { r } 
140 \
60 \
45
\end{array} \right] \
x _ { 1 } = 100 , x _ { 2 } = 200
\end{array}$$

Accepted Answer

Then @#LAT-DLM& and @#LAT-DLM& represent the "costs per doll

Question 13

Use the row reduction algorithm to transform the matrix into echelon form or reduced echelon form as indicated.
-Find the echelon form of the given matrix. $$\left[ \begin{array} { c c c c } 
1 & 4 & - 2 & 3 \
- 3 & - 11 & 9 & - 5 \
2 & 2 & 5 & - 1
\end{array} \right]$$

Accepted Answer

A) 
$\left[\begin{array}{rrrr}
1 & 4 & -2 & 3 \
0 & 1 & 3 & 4 \
0 & 0 & 15 & -1
\end{array}\right]$ 
B) 
$\left[ \begin{array} { r r r r } 1 & 4 & - 2 & 3 \ 0 & 1 & 3 & 4 \ 0 & - 6 & 9 & - 7 \end{array} \right]$ 
C)

$\left[ \begin{array} { r r r r } 1 & 4 & - 2 & 3 \ 0 & 1 & 3 & 4 \ 0 & 0 & 27 & 17 \end{array} \right]$ 
D)

$\left[ \begin{array} { r r r r } 1 & 4 & - 2 & 3 \ 0 & 1 & 3 & 4 \ 0 & 0 & 27 & 0 \end{array} \right]$ 
A) 
$\left[\begin{array}{rrrr}
1 & 4 & -2 & 3 \
0 & 1 & 3 & 4 \
0 & 0 & 15 & -1
\end{array}\right]$ 
B) 
$\left[ \begin{array} { r r r r } 1 & 4 & - 2 & 3 \ 0 & 1 & 3 & 4 \ 0 & - 6 & 9 & - 7 \end{array} \right]$ 
C)

$\left[ \begin{array} { r r r r } 1 & 4 & - 2 & 3 \ 0 & 1 & 3 & 4 \ 0 & 0 & 27 & 17 \end{array} \right]$ 
D)

$\left[ \begin{array} { r r r r } 1 & 4 & - 2 & 3 \ 0 & 1 & 3 & 4 \ 0 & 0 & 27 & 0 \end{array} \right]$

Question 14

Determine whether the system is consistent.
-$$\begin{array} { r } 
2 x _ { 1 } + x _ { 2 } = 0 \
x _ { 1 } - 3 x _ { 2 } + x _ { 3 } = 0 \
3 x _ { 1 } + x _ { 2 } - x _ { 3 } = 0
\end{array}$$

Accepted Answer

A)  $Y e s$ 
B)  No 
A)  $Y e s$ 
B)  No

Question 15

Solve the problem.
-Let $A = \left[ \begin{array} { r r r } 1 & - 3 & 2 \ - 2 & 5 & - 1 \ 3 & - 4 & 5 \end{array} \right]$ and $\mathbf { b } = \left[ \begin{array} { l } b _ { 1 } \ b _ { 2 } \ b _ { 3 } \end{array} \right]$.
Determine if the equation $\mathrm { Ax } = \mathrm { b }$ is consistent for all possible $\mathrm { b } _ { 1 } , \mathrm {~b} _ { 2 } , \mathrm {~b} _ { 3 }$. If the equation is not consistent for all possible $b _ { 1 } , b _ { 2 } , b _ { 3 }$, give a description of the set of all $\mathbf { b }$ for which the equation is consistent (i.e., a condition which must be satisfied by $b _ { 1 } , b _ { 2 } , b _ { 3 }$ ).

Accepted Answer

A)  Equation is consistent for all $b _ { 1 } , b _ { 2 } , b _ { 3 }$ satisfying $- 3 b _ { 1 } + b _ { 3 } = 0$. 
B)  Equation is consistent for all $b _ { 1 } , b _ { 2 } , b _ { 3 }$ satisfying $2 b _ { 1 } + b _ { 2 } = 0$. 
C)  Equation is consistent for all possible $b _ { 1 } , b _ { 2 } , b _ { 3 }$. 
D)  Equation is consistent for all $b _ { 1 } , b _ { 2 }$, $b _ { 3 }$ satisfying $7 b _ { 1 } + 5 b _ { 2 } + b _ { 3 } = 0$. 
A)  Equation is consistent for all $b _ { 1 } , b _ { 2 } , b _ { 3 }$ satisfying $- 3 b _ { 1 } + b _ { 3 } = 0$. 
B)  Equation is consistent for all $b _ { 1 } , b _ { 2 } , b _ { 3 }$ satisfying $2 b _ { 1 } + b _ { 2 } = 0$. 
C)  Equation is consistent for all possible $b _ { 1 } , b _ { 2 } , b _ { 3 }$. 
D)  Equation is consistent for all $b _ { 1 } , b _ { 2 }$, $b _ { 3 }$ satisfying $7 b _ { 1 } + 5 b _ { 2 } + b _ { 3 } = 0$.

Question 16

Solve the problem.
-Determine if the columns of the matrix $\mathrm { A } = \left[ \begin{array} { r r r } - 2 & 1 & 4 \ 4 & 0 & - 4 \ 2 & 1 & 0 \end{array} \right]$ are linearly independent.

Accepted Answer

A)  Yes 
B)  $\mathrm { No }$ 
A)  Yes 
B)  $\mathrm { No }$

Question 17

Find the indicated vector.
-Let $\mathbf { u } = \left[ \begin{array} { l } - 2 \ - 9 \end{array} \right] , \mathbf { v } = \left[ \begin{array} { r } 6 \ - 4 \end{array} \right]$. Find $- 2 \mathbf { u } + 5 \mathbf { v }$.

Accepted Answer

A) 
$\left[ \begin{array} { r } - 8 \ - 65 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r } - 26 \ 38 \end{array} \right]$ 
C) 
$\left[ \begin{array} { l } 22 \ 10 \end{array} \right]$ 
D) 
$\left[ \begin{array} { c } 34 \ - 2 \end{array} \right]$ 
A) 
$\left[ \begin{array} { r } - 8 \ - 65 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r } - 26 \ 38 \end{array} \right]$ 
C) 
$\left[ \begin{array} { l } 22 \ 10 \end{array} \right]$ 
D) 
$\left[ \begin{array} { c } 34 \ - 2 \end{array} \right]$

Question 18

The augmented matrix is given for a system of equations. If the system is consistent, find the general solution.
Otherwise state that there is no solution.
-$$\left[ \begin{array} { r r r r r } 
1 & 4 & - 2 & - 3 & 1 \
0 & 0 & 1 & 4 & - 4 \
- 1 & - 4 & 0 & - 5 & 7
\end{array} \right]$$

Accepted Answer

A)  $x _ { 1 } = - 4 x _ { 2 } + 2 x _ { 3 } + 3 x _ { 4 } + 1$
$x _ { 2 }$ is free
$x _ { 3 } = - 4 - 4 x _ { 4 }$
$x _ { 4 }$ is free 
B)  $\times _ { 1 } = - 7 - 4 \times _ { 2 } - 5 x _ { 4 }$
$x _ { 2 }$ is free
$x _ { 3 } = - 4 - 4 x _ { 4 }$
$x _ { 4 }$ is free 
C)  $x _ { 1 } = - 7 - 4 x _ { 2 } - 5 x _ { 4 }$
$x _ { 2 }$ is free
$x _ { 3 } = - 4 - 4 x _ { 4 }$
$x _ { 4 } = 0$ 
D)  $x _ { 1 } = - 7 - 4 x _ { 2 } - 5 \times 3$
$x _ { 2 } = - 4 - 4 x _ { 3 }$
$x _ { 3 }$ is free 
A)  $x _ { 1 } = - 4 x _ { 2 } + 2 x _ { 3 } + 3 x _ { 4 } + 1$
$x _ { 2 }$ is free
$x _ { 3 } = - 4 - 4 x _ { 4 }$
$x _ { 4 }$ is free 
B)  $\times _ { 1 } = - 7 - 4 \times _ { 2 } - 5 x _ { 4 }$
$x _ { 2 }$ is free
$x _ { 3 } = - 4 - 4 x _ { 4 }$
$x _ { 4 }$ is free 
C)  $x _ { 1 } = - 7 - 4 x _ { 2 } - 5 x _ { 4 }$
$x _ { 2 }$ is free
$x _ { 3 } = - 4 - 4 x _ { 4 }$
$x _ { 4 } = 0$ 
D)  $x _ { 1 } = - 7 - 4 x _ { 2 } - 5 \times 3$
$x _ { 2 } = - 4 - 4 x _ { 3 }$
$x _ { 3 }$ is free

Question 19

Determine whether the system is consistent.
-$$\begin{aligned}
x _ { 1 } + x _ { 2 } + x _ { 3 } & = 7 \
x _ { 1 } - x _ { 2 } + 2 x _ { 3 } & = 7 \
2 x _ { 1 } + 3 x _ { 3 } & = 15
\end{aligned}$$

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 20

Solve the problem.
-Let $\mathbf { a } _ { \mathbf { 1 } } = \left[ \begin{array} { r } 1 \ 2 \ - 3 \end{array} \right] , \mathbf { a } _ { \mathbf { 2 } } = \left[ \begin{array} { r } - 3 \ - 4 \ 1 \end{array} \right] , \mathbf { a } _ { 3 } = \left[ \begin{array} { l } 2 \ 1 \ 6 \end{array} \right]$, and $\mathbf { b } = \left[ \begin{array} { r } - 1 \ 1 \ 2 \end{array} \right]$.
Determine whether $\mathbf { b }$ can be written as a linear combination of $\mathbf { a } _ { \mathbf { 1 } } , \mathbf { a } _ { \mathbf { 2 } }$, and $\mathbf { a } _ { 3 }$. In other words, determine whether weights $x _ { 1 } , x _ { 2 }$, and $x _ { 3 }$ exist, such that $x _ { 1 } a _ { 1 } + x _ { 2 } a _ { 2 } + x _ { 3 } a _ { 3 } = b$. Determine the weights $x _ { 1 } , x _ { 2 }$, and $x _ { 3 }$ if possible.

Accepted Answer

A)  $x _ { 1 } = 2 , x _ { 2 } = 1 , x _ { 3 } = 0$ 
B)  $x _ { 1 } = - 3 , x _ { 2 } = 0 , x _ { 3 } = 1$ 
C)  No solution 
D)  $x _ { 1 } = - 2 , x _ { 2 } = - 1 , x _ { 3 } = 1$ 
A)  $x _ { 1 } = 2 , x _ { 2 } = 1 , x _ { 3 } = 0$ 
B)  $x _ { 1 } = - 3 , x _ { 2 } = 0 , x _ { 3 } = 1$ 
C)  No solution 
D)  $x _ { 1 } = - 2 , x _ { 2 } = - 1 , x _ { 3 } = 1$

Find the indicated vector. -Let $\mathbf { u } = \left[ \begin{array} { l } 6 \\ 1 \end{array} \right]$ . Find $5 \mathbf { u }$ .

Compute the product or state that it is undefined. - $\left[ \begin{array} { r r r } - 2 & - 2 & 6 \\5 & 8 & - 5\end{array} \right] \left[ \begin{array} { r } 8 \\- 3 \\3\end{array} \right]$

Find the standard matrix of the linear transformation T. -T: $\mathfrak { R } ^ { 2 } \rightarrow > \mathfrak { R } ^ { 2 }$ rotates points (about the origin) through $\frac { 7 } { 4 } \pi$ radians (with counterclockwise rotation for a positive angle).

Solve the system of equations. - 2+=0 -3+=0 3+-=0

Describe geometrically the effect of the transformation T. -Let $\mathrm { A } = \left[ \begin{array} { l l } 1 & 0 \\ 2 & 1 \end{array} \right]$ . Define a transformation $\mathrm { T }$ by $\mathrm { T } ( \mathrm { x } ) = \mathrm { Ax }$ .

Solve the system of equations. - -+8 =-107 6+ =17 3-5 =89

Determine whether the system is consistent. - 5+ =-11 ++6- =15 5++6 =16 ++3 =8

Solve the system of equations. - 5+2+=-11 2-3-=17 7++2=-4

Compute the product or state that it is undefined. - $\left[ \begin{array} { r r } 5 & - 3 \\- 3 & 4 \\- 3 & 8\end{array} \right] \left[ \begin{array} { r } 2 \\- 4 \\8\end{array} \right]$

Determine whether the system is consistent. - 5+2+=-11 2-3-=17 7++2=-4

Use the row reduction algorithm to transform the matrix into echelon form or reduced echelon form as indicated. -Find the echelon form of the given matrix. $\left[ \begin{array} { c c c c } 1 & 4 & - 2 & 3 \\- 3 & - 11 & 9 & - 5 \\2 & 2 & 5 & - 1\end{array} \right]$

Determine whether the system is consistent. - 2+=0 -3+=0 3+-=0

Solve the problem. -Determine if the columns of the matrix $\mathrm { A } = \left[ \begin{array} { r r r } - 2 & 1 & 4 \\ 4 & 0 & - 4 \\ 2 & 1 & 0 \end{array} \right]$ are linearly independent.

Find the indicated vector. -Let $\mathbf { u } = \left[ \begin{array} { l } - 2 \\ - 9 \end{array} \right] , \mathbf { v } = \left[ \begin{array} { r } 6 \\ - 4 \end{array} \right]$ . Find $- 2 \mathbf { u } + 5 \mathbf { v }$ .

The augmented matrix is given for a system of equations. If the system is consistent, find the general solution. Otherwise state that there is no solution. - $\left[ \begin{array} { r r r r r } 1 & 4 & - 2 & - 3 & 1 \\0 & 0 & 1 & 4 & - 4 \\- 1 & - 4 & 0 & - 5 & 7\end{array} \right]$

Determine whether the system is consistent. - ++ =7 -+2 =7 2+3 =15

Matrix Algebra

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Eigenvalues and Eigenvectors

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Exam 1: Linear Equations in Linear Algebra

Find the indicated vector. -Let u=[61]\mathbf { u } = \left[ \begin{array} { l } 6 \\ 1 \end{array} \right]u=[61​] . Find 5u5 \mathbf { u }5u .

Compute the product or state that it is undefined. - [−2−2658−5][8−33]\left[ \begin{array} { r r r } - 2 & - 2 & 6 \\5 & 8 & - 5\end{array} \right] \left[ \begin{array} { r } 8 \\- 3 \\3\end{array} \right][−25​−28​6−5​]​8−33​​

Find the standard matrix of the linear transformation T. -T: R2→>R2\mathfrak { R } ^ { 2 } \rightarrow > \mathfrak { R } ^ { 2 }R2→>R2 rotates points (about the origin) through 74π\frac { 7 } { 4 } \pi47​π radians (with counterclockwise rotation for a positive angle).