Consider the vectors *-(²) -(6) -(-3) = X2 X₂ = * (a) Show that X₁ and x₂ form a basis for R². (b) Why must X₁, X2, X3 be linearly dependent? (c) What is the dimension of Span(x1, X2, X3)? Given the vaatan
Consider the vectors *-(²) -(6) -(-3) = X2 X₂ = * (a) Show that X₁ and x₂ form a basis for R². (b) Why must X₁, X2, X3 be linearly dependent? (c) What is the dimension of Span(x1, X2, X3)? Given the vaatan
Advanced Engineering Mathematics
10th Edition
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Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![146 Chapter 3 Vector Spal
SECTION 3.4 EXERCISES
1. In Exercise 1 of Section 3.3, indicate whether the
given vectors form a basis for R².
2. In Exercise 2 of Section 3.3, indicate whether the
given vectors form a basis for R³.
Consider the vectors
- (1).
X₁
The standard way to represent a polynomial
1, x,x²,...,x-1, and consequently, the standard basis for Pn is {1, x,x,x).
Although these standard bases appear to be the simplest and most natural to use,
the least squares problems in Chapter 5 or the eigenvalue applications in Chapter 6.)
they are not the most appropriate bases for many applied problems. (See, for example,
Indeed, the key to solving many applied problems is to switch from one of the standard
bases to a basis that is in some sense natural for the particular application. Once the
application is solved in terms of the new basis, it is a simple matter to switch back and
represent the solution in terms of the standard basis. In the next section, we will learn
how to switch from one basis to another.
X₁ =
X₂ =
(a) Show that x₁ and x₂ form a basis for R².
(b) Why must X₁, X2, X3 be linearly dependent?
(c) What is the dimension of Span(x1, X2, X3)?
Given the vectors
(3) ×₁=(-3)
X3
X₂ =
X3 =
4
-8
what is the dimension of Span(X₁, X2, X3)?
5. Let
11 Jnsbro
3
*- *- --B
=
X3 =
(a) Show that X₁, X2, and x3 are linearly dependent.
(b) Show that x, and x₂ are linearly independent.
(c) What is the dimension of Span(x₁, X2, X3)?
(d) Give a geometric description
Span(X1, X2, X3).
6. In Exercise 2 of Section 3.2, some of the sets
formed subspaces of R³. In each of these cases,
find a basis for the subspace and determine its
dimension.
of
Find a basis for the subspace S of R4 consisting of
all vectors of the form (a + b,a - b + 2c, b, c),
where a, b, and c are all real numbers. What is the
dimension of S?
8. Given x₁ = (1, 1, 1) and x₂ = (3, -1, 4)T:
(a) Do x₁ and x₂ span R³? Explain.
(b) Let x3 be a third vector in R³ and set X-
(X₁ X2 X3). What condition(s) would X have
to satisfy in order for X₁, X2, and x3 to form a
basis for R³?
(c) Find a third vector x3 that will extend the set
(X₁, X2) to a basis for R³.
9. Let a, and a2 be linearly independent vectors in R³,
and let x be a vector in R².
(a) Describe geometrically Span(a₁, a₂).
(b) If A = (a₁, a₂) and b = Ax, then what is the
dimension of Span(a₁, a2, b)? Explain.
10. The vectors
X₁ =
X3 =
(3).
tions
, X₂ =
----
span R³. Pare down the set {x1, X2, X3, X4, X5} to
form a basis for R³.
, X4=
=
Let S be the subspace of P3 consisting of all polyno-
mials of the form ax² + bx + 2a + 3b. Find a basis
for S.
12. In Exercise 3 of Section 3.2, some of the sets
formed subspaces of R2x2. In each of these cases,
find a basis for the subspace and determine its
dimension.
13. In C[-7,7], find the dimension of the subspace
spanned by 1, cos 2x, cos² x.
14. In each of the following, find the dimension of the
subspace of P3 spanned by the given vectors:
(a) x,x-1,x² +1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe0357f25-03e2-4d9e-b3d6-59a44a42e09c%2Ff4e2bd51-f2ce-4885-bff8-8ee0ec457db2%2F0993tjh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:146 Chapter 3 Vector Spal
SECTION 3.4 EXERCISES
1. In Exercise 1 of Section 3.3, indicate whether the
given vectors form a basis for R².
2. In Exercise 2 of Section 3.3, indicate whether the
given vectors form a basis for R³.
Consider the vectors
- (1).
X₁
The standard way to represent a polynomial
1, x,x²,...,x-1, and consequently, the standard basis for Pn is {1, x,x,x).
Although these standard bases appear to be the simplest and most natural to use,
the least squares problems in Chapter 5 or the eigenvalue applications in Chapter 6.)
they are not the most appropriate bases for many applied problems. (See, for example,
Indeed, the key to solving many applied problems is to switch from one of the standard
bases to a basis that is in some sense natural for the particular application. Once the
application is solved in terms of the new basis, it is a simple matter to switch back and
represent the solution in terms of the standard basis. In the next section, we will learn
how to switch from one basis to another.
X₁ =
X₂ =
(a) Show that x₁ and x₂ form a basis for R².
(b) Why must X₁, X2, X3 be linearly dependent?
(c) What is the dimension of Span(x1, X2, X3)?
Given the vectors
(3) ×₁=(-3)
X3
X₂ =
X3 =
4
-8
what is the dimension of Span(X₁, X2, X3)?
5. Let
11 Jnsbro
3
*- *- --B
=
X3 =
(a) Show that X₁, X2, and x3 are linearly dependent.
(b) Show that x, and x₂ are linearly independent.
(c) What is the dimension of Span(x₁, X2, X3)?
(d) Give a geometric description
Span(X1, X2, X3).
6. In Exercise 2 of Section 3.2, some of the sets
formed subspaces of R³. In each of these cases,
find a basis for the subspace and determine its
dimension.
of
Find a basis for the subspace S of R4 consisting of
all vectors of the form (a + b,a - b + 2c, b, c),
where a, b, and c are all real numbers. What is the
dimension of S?
8. Given x₁ = (1, 1, 1) and x₂ = (3, -1, 4)T:
(a) Do x₁ and x₂ span R³? Explain.
(b) Let x3 be a third vector in R³ and set X-
(X₁ X2 X3). What condition(s) would X have
to satisfy in order for X₁, X2, and x3 to form a
basis for R³?
(c) Find a third vector x3 that will extend the set
(X₁, X2) to a basis for R³.
9. Let a, and a2 be linearly independent vectors in R³,
and let x be a vector in R².
(a) Describe geometrically Span(a₁, a₂).
(b) If A = (a₁, a₂) and b = Ax, then what is the
dimension of Span(a₁, a2, b)? Explain.
10. The vectors
X₁ =
X3 =
(3).
tions
, X₂ =
----
span R³. Pare down the set {x1, X2, X3, X4, X5} to
form a basis for R³.
, X4=
=
Let S be the subspace of P3 consisting of all polyno-
mials of the form ax² + bx + 2a + 3b. Find a basis
for S.
12. In Exercise 3 of Section 3.2, some of the sets
formed subspaces of R2x2. In each of these cases,
find a basis for the subspace and determine its
dimension.
13. In C[-7,7], find the dimension of the subspace
spanned by 1, cos 2x, cos² x.
14. In each of the following, find the dimension of the
subspace of P3 spanned by the given vectors:
(a) x,x-1,x² +1
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