23. The accompanying figure shows a rectangular xy-coordin- ate system determined by the unit basis vectors i and j and an x'y-coordinate system determined by unit basis vectors u, and u₂. Find the x'y'-coordinates of the points whose xy-coordinates are given. a. (√3,1) b. (1,0) c. (0,1) d. (a,b) Ay and y' X jand u₂ 30° u₁
23. The accompanying figure shows a rectangular xy-coordin- ate system determined by the unit basis vectors i and j and an x'y-coordinate system determined by unit basis vectors u, and u₂. Find the x'y'-coordinates of the points whose xy-coordinates are given. a. (√3,1) b. (1,0) c. (0,1) d. (a,b) Ay and y' X jand u₂ 30° u₁
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 76E
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23. The accompanying figure shows a rectangular xy-coordin-
ate system determined by the unit basis vectors i and j and
an x'y'-coordinate system determined by unit basis vectors
u₁ and u₂. Find the x'y'-coordinates of the points whose
xy-coordinates are given.
a. (√3,1)
b. (1,0)
c. (0,1) d. (a,b)
Ay and y'
j and u₂
30°
x
FIGURE Ex-23
24. The accompanying figure shows a rectangular xy-coordinate
system and an x'y'-coordinate system with skewed axes.
Assuming that 1-unit scales are used on all the axes, find the
x'y'-coordinates of the points whose xy-coordinates are given.
a. (1,1) b. (1,0) c. (0,1) d. (a, b)
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u₁
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4.5 Coordinates and Basis 247
a. Find w if S is the basis in Exercise 2.
b. Find q if S is the basis in Exercise 3.
c. Find B if S is the basis in Exercise 5.
28. The basis that we gave for M₂2 in Example 4 consisted of non-
invertible matrices. Do you think that there is a basis for M22
consisting of invertible matrices? Justify your answer.
Working with Proofs
29. Prove that R* is an infinite-dimensional vector space.
30. Let TARR" be multiplication by an invertible matrix
A, and let {u₁, ₂,..., u} be a basis for R". Prove that
{TA(U₁), TA (U₂),..., TA (un)} is also a basis for R".
31. Prove that if V is a subspace of a vector space W and if V is
infinite-dimensional, then so is W.
True-False Exercises
TF. In parts (a)-(e) determine whether the statement is true or
false, and justify your answer.
a. If V = span{v₁,...,Vn}, then {v₁,...,v} is a basis for V.
b. Every linearly independent subset of a vector space V is a
basis for V.
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□ O
ΣΣΣΣΣΣ
A+ A
Cover
Applications and Historical
Topics
Title Page
Copyright
About the Authors
Preface
Contents
> 1: Systems of Linear Equations
and Matrices
>
2: Determinants
>
3: Euclidean Vector Spaces
>4: General Vector Spaces
> 5: Eigenvalues and
Eigenvectors
>
6: Inner Product Spaces
7: Diagonalization and
Quadratic Forms
Add Image Link
>
> 8: General Linear
Transformations
>9: Numerical Methods
Appendix A: Working with
Proofs
se
Elementary_Linear_Algebra_by_Howard_Anton, Anton_Kaul_z_lib_org.pdf
Comment Edit Convert Tool View Form Protect Page Template
Watermark
Background
In
Header & Footer 1 Bates Number
At least one signature has problems.
23. The accompanying figure shows a rectangular xy-coordin-
ate system determined by the unit basis vectors i and j and
an x'y'-coordinate system determined by unit basis vectors
u₁ and u₂. Find the x'y'-coordinates of the points whose
xy-coordinates are given.
a. (√3,1)
b. (1,0)
c. (0,1) d. (a,b)
Ay and y'
j and u₂
30°
x
FIGURE Ex-23
24. The accompanying figure shows a rectangular xy-coordinate
system and an x'y'-coordinate system with skewed axes.
Assuming that 1-unit scales are used on all the axes, find the
x'y'-coordinates of the points whose xy-coordinates are given.
a. (1,1) b. (1,0) c. (0,1) d. (a, b)
APO
u₁
iPhone/iPad
Buy Now
Edit Mode
4.5 Coordinates and Basis 247
a. Find w if S is the basis in Exercise 2.
b. Find q if S is the basis in Exercise 3.
c. Find B if S is the basis in Exercise 5.
28. The basis that we gave for M₂2 in Example 4 consisted of non-
invertible matrices. Do you think that there is a basis for M22
consisting of invertible matrices? Justify your answer.
Working with Proofs
29. Prove that R* is an infinite-dimensional vector space.
30. Let TARR" be multiplication by an invertible matrix
A, and let {u₁, ₂,..., u} be a basis for R". Prove that
{TA(U₁), TA (U₂),..., TA (un)} is also a basis for R".
31. Prove that if V is a subspace of a vector space W and if V is
infinite-dimensional, then so is W.
True-False Exercises
TF. In parts (a)-(e) determine whether the statement is true or
false, and justify your answer.
a. If V = span{v₁,...,Vn}, then {v₁,...,v} is a basis for V.
b. Every linearly independent subset of a vector space V is a
basis for V.
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