Ix F-CH+CH
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
5
![32 CHAPTER 1 Linear Equations in Linear Algebra
nort
A
worairt nowns of noitulos e
PRACTICE PROBLEMS
1. Prove that u + v = v +u for any u and v in R".
omsiq Jam 2. For what value(s) of h will y be in Span{V₁, V2, V3} if
fonsig
5
-4
--------
toloov oil noiioz on 250
1.3 EXERCISES
5. X₁
d
H
6
5
bas
2. u = [2] . v = [-²]
In Exercises 3 and 4, display the following vectors using arrows
amun
on an xy-graph: u, v, -v, -2v, u + v, u-v, and u-2v. Notice
STILL.CO
that u - v is the vertex of a parallelogram whose other vertices are
u, 0, and -v.
0
1. u=
= [²1] = []
mar 200 169nil bas asigillum als word 29. le olqmX₂ + 5x3 = 0
³] manif bris
nov.0
161
3. u and v as in Exercise 1 4. u and v as in Exercise 2
sachsvo no clcbus,nos no CS. lsdotén
d
ated by
In Exercises 9 and 10, write a vector equation that is equivalent to
In Exercises 1 and 2, compute u + v and u-2v.
anonsonqgA ni enol the given system of equations.
HUB
=
+ x2
0
-2v
a
1.5
7. Vectors a, b, c, and d
2010
rol
=
*[3] + [3] + [6] - [8]
6. X1
x2
x3
-6
27pogolo Isrovni awobrisdi
lo alinu Imoves gniouborq to 1200 od among r
b
-V
-U
H
=
-5
0001 10100
SU
Use the accompanying figure to write each vector listed in Exer-
cises 7 and 8 as a linear combination of u and v. Is every vector
1
in R2 a linear combination of u and v? TOY SVIO
(bascovo bigod
3. Let W1, W2, W3, u, and v be vectors in R". Suppose the vectors u and v are in Span
{W1, W2, W3}. Show that u + v is also in Span (W₁, W2, W3}. [Hint: The solution to
Practice Problem 3 requires the use of the definition of the span of a set of vectors.
It is useful to review this definition on Page 30 before starting this exercise.]
arT
terü avrora dordy
noitrups
Une combl
In Exercises 5 and 6, write a system of equations that is equivalent noo, oh Odoubring todizovacala
to the given vector equation.
ple), since
1
V CA
X
W
2v
VA
=
y
-
1
011001
1²/
Z 18V 1
252falsi
novi s to drown
lo
8. Vectors w, x, y, and zdolay sill
20
4x1 + 6x2 - x3 = 0
-x₁ + 3x₂ - 8x3 = 0
21 ling
11. a₁ =
=
V3
In Exercises 11 and 12, determine if b is a linear combination of
a1, a2, and a3.
0
, =
------
2
12. a₁ =
1
-2
14. A =
-2
15. V₁ =
=> bos
= d
2
Geometric Desc In Exercises 13 and 14, determine if b is a linear combination of
which
the vectors formed from the columns of the matrix A.
-1
13. A =
tistimos-2008
10 dnd
Fur
=
0
, = 5 , =
--------
5
1 -4 2
0 3 5 ,b=
1-2-6
3
0
14 -2 5
[1]
-6
370
16. V₁ =
0
uteng 2
and y =
10. 4x₁ + x2 + 3x3 = 9
x₁ - 7x₂ - 2x3 = 2
8x₁ + 6x₂ - 5x3 = 15
27.0²
b
3]-[
-7
, V₂ =
, V2: =
0
3
-6
2
11
- -5
9
5
3
0
3
h
8
8
In Exercises 15 and 16, list five vectors in Span {V₁, V₂). For each
vector, show the weights on v₁ and v2 used to generate the vector
and list the three entries of the vector. Do not make a sketch.
, b =
-1
-[-²]
6
b= 11
17](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9aabfb62-88cc-47fc-a128-866ddac4a30f%2F3d43a62e-cf71-4338-87bc-1e24db28e34c%2F8ha1kpi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:32 CHAPTER 1 Linear Equations in Linear Algebra
nort
A
worairt nowns of noitulos e
PRACTICE PROBLEMS
1. Prove that u + v = v +u for any u and v in R".
omsiq Jam 2. For what value(s) of h will y be in Span{V₁, V2, V3} if
fonsig
5
-4
--------
toloov oil noiioz on 250
1.3 EXERCISES
5. X₁
d
H
6
5
bas
2. u = [2] . v = [-²]
In Exercises 3 and 4, display the following vectors using arrows
amun
on an xy-graph: u, v, -v, -2v, u + v, u-v, and u-2v. Notice
STILL.CO
that u - v is the vertex of a parallelogram whose other vertices are
u, 0, and -v.
0
1. u=
= [²1] = []
mar 200 169nil bas asigillum als word 29. le olqmX₂ + 5x3 = 0
³] manif bris
nov.0
161
3. u and v as in Exercise 1 4. u and v as in Exercise 2
sachsvo no clcbus,nos no CS. lsdotén
d
ated by
In Exercises 9 and 10, write a vector equation that is equivalent to
In Exercises 1 and 2, compute u + v and u-2v.
anonsonqgA ni enol the given system of equations.
HUB
=
+ x2
0
-2v
a
1.5
7. Vectors a, b, c, and d
2010
rol
=
*[3] + [3] + [6] - [8]
6. X1
x2
x3
-6
27pogolo Isrovni awobrisdi
lo alinu Imoves gniouborq to 1200 od among r
b
-V
-U
H
=
-5
0001 10100
SU
Use the accompanying figure to write each vector listed in Exer-
cises 7 and 8 as a linear combination of u and v. Is every vector
1
in R2 a linear combination of u and v? TOY SVIO
(bascovo bigod
3. Let W1, W2, W3, u, and v be vectors in R". Suppose the vectors u and v are in Span
{W1, W2, W3}. Show that u + v is also in Span (W₁, W2, W3}. [Hint: The solution to
Practice Problem 3 requires the use of the definition of the span of a set of vectors.
It is useful to review this definition on Page 30 before starting this exercise.]
arT
terü avrora dordy
noitrups
Une combl
In Exercises 5 and 6, write a system of equations that is equivalent noo, oh Odoubring todizovacala
to the given vector equation.
ple), since
1
V CA
X
W
2v
VA
=
y
-
1
011001
1²/
Z 18V 1
252falsi
novi s to drown
lo
8. Vectors w, x, y, and zdolay sill
20
4x1 + 6x2 - x3 = 0
-x₁ + 3x₂ - 8x3 = 0
21 ling
11. a₁ =
=
V3
In Exercises 11 and 12, determine if b is a linear combination of
a1, a2, and a3.
0
, =
------
2
12. a₁ =
1
-2
14. A =
-2
15. V₁ =
=> bos
= d
2
Geometric Desc In Exercises 13 and 14, determine if b is a linear combination of
which
the vectors formed from the columns of the matrix A.
-1
13. A =
tistimos-2008
10 dnd
Fur
=
0
, = 5 , =
--------
5
1 -4 2
0 3 5 ,b=
1-2-6
3
0
14 -2 5
[1]
-6
370
16. V₁ =
0
uteng 2
and y =
10. 4x₁ + x2 + 3x3 = 9
x₁ - 7x₂ - 2x3 = 2
8x₁ + 6x₂ - 5x3 = 15
27.0²
b
3]-[
-7
, V₂ =
, V2: =
0
3
-6
2
11
- -5
9
5
3
0
3
h
8
8
In Exercises 15 and 16, list five vectors in Span {V₁, V₂). For each
vector, show the weights on v₁ and v2 used to generate the vector
and list the three entries of the vector. Do not make a sketch.
, b =
-1
-[-²]
6
b= 11
17
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