A.7 Exercises
Your Turn
On a piece of graph paper draw the vector ( 1 , 2 )
based at ( 0 , 0 ) based at ( 1 , 2 ) based at ( 0 , − 1 ) Your Turn
On a piece of graph paper draw the following operations. Draw and label the vectors involved in the operations as well as the result:
[
1
−
4
]
+
[
2
3
]
[
−
3
2
]
−
[
1
3
]
3
[
2
1
]
Your Turn
Compute the magnitude of
[
7
2
]
[
−
2
3
1
]
(
1
,
3
,
−
4
)
Your Turn
Compute
[
2
3
]
+
[
7
−
8
]
[
−
2
3
]
−
[
6
−
4
]
−
[
−
3
2
]
4
[
−
1
5
]
5
[
1
0
]
+
9
[
0
1
]
3
[
1
−
8
]
−
2
[
3
−
1
]
Your Turn
If x → = ( 1 , 2 ) y → x → + y → = ( 0 , 2 ) y →
Your Turn
Write ( 1 , 2 , 3 ) e → 1 e → 2 e → 3
Your Turn
If the magnitude of x →
0
x
→
3
x
→
−
x
→
−
4
x
→
x
→
+
x
→
x
→
−
x
→
Your Turn
Suppose a linear mapping F : R 2 → R 2 ( 1 , 0 ) ( 2 , − 1 ) ( 0 , 1 ) ( 3 , 3 )
(
1
,
1
)
(
2
,
0
)
(
2
,
−
1
)
Your Turn
Suppose a linear mapping F : R 3 → R 2 ( 1 , 0 , 0 ) ( 2 , 1 ) ( 0 , 1 , 0 ) ( 3 , 4 ) ( 0 , 0 , 1 ) ( 5 , 6 ) F
Your Turn
Suppose that a mapping F : R 2 → ℝ 2 ( 1 , 0 ) ( 1 , 2 ) ( 0 , 1 ) ( 3 , 4 ) ( 1 , 1 ) ( 0 , − 1 ) F
Your Turn
(challenging)
Let R 3 t ( a 0 , a 1 , a 2 ) R 3 a 0 + a 1 t + a 2 t 2 d d t R 3 R 3 d d t d d t 3 × 3
Your Turn
Compute the magnitude of
[
1
3
]
[
2
3
−
1
]
(
−
2
,
1
,
−
2
)
Answer
10
14
3
Your Turn
Compute
[
3
1
]
+
[
6
−
3
]
[
−
1
2
]
−
[
2
−
1
]
−
[
−
5
3
]
2
[
−
2
4
]
3
[
1
0
]
+
7
[
0
1
]
2
[
2
−
3
]
−
6
[
2
−
1
]
Answer
9
−
2
−
3
3
5
−
3
−
4
8
3
7
−
8
3
Your Turn
If the magnitude of x →
4
x
→
−
2
x
→
−
4
x
→
Answer
20
10
20
Your Turn
Suppose a linear mapping F : R 2 → R 2 ( 1 , 0 ) ( 1 , − 1 ) ( 0 , 1 ) ( 2 , 0 )
(
1
,
1
)
(
0
,
2
)
(
1
,
−
1
)
Answer
(
3
,
−
1
)
(
4
,
0
)
(
−
1
,
−
1
)
Your Turn
Compute
3
[
0
3
−
2
2
]
+
6
[
1
5
−
1
5
]
2
[
−
3
1
2
2
]
−
3
[
2
−
1
3
2
]
Your Turn
Add the following matrices
[
2
1
0
1
1
−
1
]
+
[
5
3
4
1
2
5
]
[
6
−
2
3
7
3
3
8
−
1
2
]
+
[
−
1
−
1
−
3
6
7
3
−
9
4
−
1
]
Answer
Add texts here. Do not delete this text first.
Your Turn
Multiply the following matrices
[
2
1
4
3
4
4
]
[
2
4
6
3
3
5
]
[
0
3
3
2
−
2
1
3
5
−
2
]
[
6
6
2
4
6
0
2
0
4
]
[
3
4
1
2
−
1
0
4
−
1
5
]
[
0
2
5
0
2
0
5
2
3
6
1
6
]
[
−
2
−
2
5
3
2
1
]
[
0
3
1
3
]
Answer
Add texts here. Do not delete this text first.
Your Turn
Compute the inverse of the given matrices
[
2
]
[
0
1
1
0
]
[
1
2
3
5
]
[
4
2
4
4
]
Answer
Add texts here. Do not delete this text first.
Your Turn
Compute the inverse of the given matrices
[
2
0
0
3
]
[
4
0
0
0
5
0
0
0
−
1
]
[
−
1
0
0
0
0
2
0
0
0
0
3
0
0
0
0
0.1
]
Answer
Add texts here. Do not delete this text first.
Your Turn
Solve (find all solutions), or show no solution exists
4
x
1
+
3
x
2
=
−
2
−
x
1
+
x
2
=
4
x
1
+
5
x
2
+
3
x
3
=
7
8
x
1
+
7
x
2
+
8
x
3
=
8
4
x
1
+
8
x
2
+
6
x
3
=
4
4
x
1
+
8
x
2
+
2
x
3
=
3
−
x
1
−
2
x
2
+
3
x
3
=
1
4
x
1
+
8
x
2
+
3
x
3
}
=
2
x
+
2
y
+
3
z
=
4
2
x
−
y
+
3
z
=
1
3
x
+
y
+
6
z
=
6
Your Turn
By computing the inverse, solve the following systems for x →
[
4
1
−
1
3
]
x
→
=
[
13
26
]
[
3
3
3
4
]
x
→
=
[
2
−
1
]
Your Turn
For the matrices in Exercise A .3 .5
Your Turn
For the matrices in Exercise A .3 .5
Your Turn
Find a linearly independent subset of the following vectors that has the same span.
[ − 1 1 2 ] , [ 2 − 2 − 4 ] , [ − 2 4 1 ] , [ − 1 3 − 2 ]
Your Turn
Solve (find all solutions), or show no solution exists
4
x
1
+
3
x
2
=
−
1
5
x
1
+
6
x
2
=
4
5
x
+
6
y
+
5
z
=
7
6
x
+
8
y
+
6
z
=
−
1
5
x
+
2
y
+
5
z
=
2
a
+
b
+
c
=
−
1
a
+
5
b
+
6
c
=
−
1
−
2
a
+
5
b
+
6
c
=
8
−
2
x
1
+
2
x
2
+
8
x
3
=
6
x
2
+
x
3
=
2
x
1
+
4
x
2
+
x
3
=
7
Answer
x
1
=
−
2
,
x
2
=
7
3
no solution
a
=
−
3
,
b
=
10
,
c
=
−
8
x 3 x 1 = − 1 + 3 x 3 x 2 = 2 − x 3 Your Turn
By computing the inverse, solve the following systems for x →
[
−
1
1
3
3
]
x
→
=
[
4
6
]
[
2
7
1
6
]
x
→
=
[
1
3
]
Answer
−
1
3
−
3
1
Your Turn
For the matrices in Exercise A .3 .13
Answer
1 0 0 0 1 0 0 0 1
1
1
1
1 0 1 3 0 1 − 1 3 Your Turn
For the matrices in Exercise A .3 .13
Answer
7 7 7 − 1 7 6 7 6 2
1
1
2
0 6 4 3 3 7 Your Turn
Find a linearly independent subset of the following vectors that has the same span.
[ 0 0 0 ] , [ 3 1 − 5 ] , [ 0 3 − 1 ] , [ − 3 2 4 ]
Answer
3 1 − 5 0 3 − 1
Your Turn
For the following sets of vectors, find a basis for the subspace spanned by the vectors, and find the dimension of the subspace.
[
1
1
1
]
,
[
−
1
−
1
−
1
]
[
1
0
5
]
,
[
0
1
0
]
,
[
0
−
1
0
]
[
−
4
−
3
5
]
,
[
2
3
3
]
,
[
2
0
2
]
[
1
3
0
]
,
[
0
2
2
]
,
[
−
1
−
1
2
]
[
1
3
]
,
[
0
2
]
,
[
−
1
−
1
]
[
3
1
3
]
,
[
2
4
−
4
]
,
[
−
5
−
5
−
2
]
Your Turn
For the following matrices, find a basis for the kernel (nullspace).
[
1
1
1
1
1
5
1
1
−
4
]
[
2
−
1
−
3
4
0
−
4
−
1
1
2
]
[
−
4
4
4
−
1
1
1
−
5
5
5
]
[
−
2
1
1
1
−
4
2
2
2
1
0
4
3
]
Your Turn
Suppose that X ℝ 3 X
Your Turn
Consider a square matrix A x → A x → = 0 → A
Your Turn
Consider
M = [ 1 2 3 2 ? ? − 1 ? ? ] .
If the nullity of this matrix is 2, fill in the question marks. Hint: What is the rank?
Your Turn
For the following sets of vectors, find a basis for the subspace spanned by the vectors, and find the dimension of the subspace.
[
1
2
]
,
[
1
1
]
[
1
1
1
]
,
[
2
2
2
]
,
[
1
1
2
]
[
5
3
1
]
,
[
5
−
1
5
]
,
[
−
1
3
−
4
]
[
2
2
4
]
,
[
2
2
3
]
,
[
4
4
−
3
]
[
1
0
]
,
[
2
0
]
,
[
3
0
]
[
1
0
0
]
,
[
2
0
0
]
,
[
0
1
2
]
Answer
1 2 1 1 2 1 1 1 1 1 2 2 5 3 1 5 − 1 5 − 1 3 − 4 3 2 2 4 2 2 3 2 1 1 1 1 0 0 0 1 2 2 Your Turn
For the following matrices, find a basis for the kernel (nullspace).
[
2
6
1
9
1
3
2
9
3
9
0
9
]
[
2
−
2
−
5
−
1
1
5
−
5
5
−
3
]
[
1
−
5
−
4
2
3
5
−
3
5
2
]
[
0
4
4
0
1
1
0
5
5
]
Answer
3 − 1 0 0 3 0 3 − 1
−
1
−
1
0
1
1
−
1
− 1 0 0 0 1 − 1 Your Turn
Find the s ( 1 , 2 , 3 ) ( 1 , 1 , s )
Your Turn
Find the angle θ ( 1 , 3 , 1 ) ( 2 , 1 , − 1 )
Your Turn
Given that ⟨ v → , w → ⟩ = 3 ⟨ v → , u → ⟩ = − 1
⟨
u
→
,
2
v
→
⟩
⟨
v
→
,
2
w
→
+
3
u
→
⟩
⟨
w
→
+
3
u
→
,
v
→
⟩
Your Turn
Suppose v → = ( 1 , 1 , − 1 )
proj
v
→
(
(
1
,
0
,
0
)
)
proj
v
→
(
(
1
,
2
,
3
)
)
proj
v
→
(
(
1
,
−
1
,
0
)
)
Your Turn
Consider the vectors ( 1 , 2 , 3 ) ( − 3 , 0 , 1 ) ( 1 , − 5 , 3 )
Check that the vectors are linearly independent and so form a basis. Check that the vectors are mutually orthogonal, and are therefore an orthogonal basis. Represent ( 1 , 1 , 1 ) Make the basis orthonormal. Your Turn
Let S ( 1 , 3 , − 1 ) ( 1 , 1 , 1 ) S
Your Turn
Starting with ( 1 , 2 , 3 ) ( 1 , 1 , 1 ) ( 2 , 2 , 0 ) ℝ 3
Your Turn
Find an orthogonal basis of ℝ 3 ( 3 , 1 , − 2 )
Your Turn
Using cosines and sines of θ u → ℝ 2 θ ı → = ( 1 , 0 ) ⟨ ı → , u → ⟩
Your Turn
Find the s ( 1 , 1 , 1 ) ( 1 , s , 1 )
Answer
s
=
−
2
Your Turn
Find the angle θ ( 1 , 2 , 3 ) ( 1 , 1 , 1 )
Answer
θ
≈
0.3876
Your Turn
Given that ⟨ v → , w → ⟩ = 1 ⟨ v → , u → ⟩ = − 1 ‖ v → ‖ = 3
⟨
3
u
→
,
5
v
→
⟩
⟨
v
→
,
2
w
→
+
3
u
→
⟩
⟨
w
→
+
3
v
→
,
v
→
⟩
Answer
−
15
−
1
28
Your Turn
Suppose v → = ( 1 , 0 , − 1 )
proj
v
→
(
(
0
,
2
,
1
)
)
proj
v
→
(
(
1
,
0
,
1
)
)
proj
v
→
(
(
4
,
−
1
,
0
)
)
Answer
(
−
1
2
,
0
,
1
2
)
(
0
,
0
,
0
)
(
2
,
0
,
−
2
)
Your Turn
The vectors ( 1 , 1 , − 1 ) ( 2 , − 1 , 1 ) ( 1 , − 5 , 3 )
(
1
,
−
8
,
4
)
(
5
,
−
7
,
5
)
(
0
,
−
6
,
2
)
Answer
(
1
,
1
,
−
1
)
−
(
2
,
−
1
,
1
)
+
2
(
1
,
−
5
,
3
)
2
(
2
,
−
1
,
1
)
+
(
1
,
−
5
,
3
)
2
(
1
,
1
,
−
1
)
−
2
(
2
,
−
1
,
1
)
+
2
(
1
,
−
5
,
3
)
Your Turn
Let S ( 2 , − 1 , 1 ) ( 2 , 2 , 2 ) S
Answer
( 2 , − 1 , 1 ) ( 2 3 , 8 3 , 4 3 )
Your Turn
Starting with ( 1 , 1 , − 1 ) ( 2 , 3 , − 1 ) ( 1 , − 1 , 1 ) ℝ 3
Answer
( 1 , 1 , − 1 ) ( 0 , 1 , 1 ) ( 4 3 , − 2 3 , 2 3 )
Your Turn
Consider the linear mapping from R 2 R 2 A = [ 1 x 2 1 ] x A x
Your Turn
Suppose A S n × n S det ( A ) = 3 det ( S − 1 A S ) det ( S A S − 1 )
Your Turn
Let A n × n det ( A ) = 1 det ( x A ) x det ( x I ) x A = ( x I ) A
Adapted from Differential Equations for Engineers by Jiří Lebl (Oklahoma State University), hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0 .
