In Exercises 13 and 14, determine if b is a linear combination of the vectors formed from the columns of the matrix A. r- 13. A = or vint 1 -4 2 0 3 5 -2 8-4 [2 -1- 3 -7 -3 .b= contains thens

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10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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
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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