Problem 1E W is a subset of R2 consisting of vectors of the form To determine whether W is a subspace of R2. If W is a subspace, then give a geometric description of W. W = {x: x1 = 2x2}
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A.1
A.10
A.2
A.3
A.4
A.6
A.7
A.8
A.9
1.1
Introduction to Matrices and Systems of Linear Equations
1.2
Echelon Form and Gauss–Jordan Elimination
1.3
Consistent Systems of Linear Equations
1.4
Applications (Optional)
1.5
Matrix Operations
1.6
In this discussion of matrices and matrix operations (and later in the discussion of
1.7
vectors), it is customary to refer to numerical quantities as scalars.
1.8
Data Fitting, Numerical Integration, and Numerical Differentiation (Optional)
1.9
Matrix Inverses and Their Properties
2.1
Vectors in the Plane
2.2
Vectors in Space
3.1
Introduction
3.2
Vector Space Properties of Rn
3.8
Least-Squares Solutions to Inconsistent Systems, with Applications to Data Fitting
3.9
Theory and Practice of Least Squares
4.1
The Eigenvalue Problem for (2 × 2) Matrices
4.2
Determinants and the Eigenvalue Problem
Textbook Solutions for Introduction to Linear Algebra
Chapter 3.2 Problem 32E
Question
Let U and V be the subspaces of R3 defined byU = {x: aT x = 0} and V = {x: bT x = 0}Where Demonstrate that the union, U ? V, is not a subspace of R3(see Let abe a fixed vector in R3, and define W to be the subset of R3 given by w = [x: aTx = 0] Prove that W is a subspace ofR3)
Solution
The first step in solving 3.2 problem number 32 trying to solve the problem we have to refer to the textbook question: Let U and V be the subspaces of R3 defined byU = {x: aT x = 0} and V = {x: bT x = 0}Where Demonstrate that the union, U ? V, is not a subspace of R3(see Let abe a fixed vector in R3, and define W to be the subset of R3 given by w = [x: aTx = 0] Prove that W is a subspace ofR3)
From the textbook chapter Vector Space Properties of Rn you will find a few key concepts needed to solve this.
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full solution
full solution
Title
Introduction to Linear Algebra 5
Author
Lee W. Johnson, R. Dean Riess, Jimmy T. Arnold
ISBN
9780201658590
Let U and V be the subspaces of R3 defined byU = {x: aT x
Chapter 3.2 textbook questions
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Chapter 3: Problem 1 Introduction to Linear Algebra 5
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Chapter 3: Problem 4 Introduction to Linear Algebra 5
Problem 4E W is a subset of R2 consisting of vectors of the form To determine whether W is a sub space of R2. If W is a subspace, then give a geometric description of W. W = {x: x1 and x2 are x2 are rational numbers}
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Chapter 3: Problem 34 Introduction to Linear Algebra 5
Problem 34E Let W be a nonempty subset of Rn that satisfies conditions (s2) and (s3) of Theorem 2. Prove that ? is in W and conclude that VV is a subspace of Rn. (Thus property (si) of Theorem 2 can be replaced with the assumption that W is nonempty.)
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Chapter 3: Problem 2 Introduction to Linear Algebra 5
Problem 2E W is a subset of R2 consisting of vectors of the form To determine whether W is a sub space of R2. If W is a subspace, then give a geometric description of W. W = {x: x1 – x2 = 2}
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Chapter 3: Problem 3 Introduction to Linear Algebra 5
Problem 3E W is a subset of R2 consisting of vectors of the form To determine whether W is a sub space of R2. If W is a subspace, then give a geometric description of W. W = {x: x1 = x2 or x1 = -x2}
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Chapter 3: Problem 5 Introduction to Linear Algebra 5
Problem 5E W is a subset of R2 consisting of vectors of the form To determine whether W is a sub space of R2. If W is a subspace, then give a geometric description of W. W = {x: x1 = 0}
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Chapter 3: Problem 6 Introduction to Linear Algebra 5
Problem 6E W is a subset of R2 consisting of vectors of the form To determine whether W is a sub space of R2. If W is a subspace, then give a geometric description of W. W = {x: |x1| + |x2| = 0}
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Chapter 3: Problem 9 Introduction to Linear Algebra 5
Problem 9E W is a subset of R3 consisting of vectors of the form Determine whether W is a subspace of R3 If W is a subspace, then give a geometric description of W. W = {x: x3 = 2x1 - x2}
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Chapter 3: Problem 8 Introduction to Linear Algebra 5
Problem 8E W is a subset of R2 consisting of vectors of the form To determine whether W is a sub space of R2. If W is a subspace, then give a geometric description of W. W = {x: x1 x2 = 0}
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Chapter 3: Problem 10 Introduction to Linear Algebra 5
Problem 10E W is a subset of R3 consisting of vectors of the form Determine whether W is a subspace of R3 If W is a subspace, then give a geometric description of W. W = {x: x2 = x3 + x1}
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Chapter 3: Problem 7 Introduction to Linear Algebra 5
Problem 7E W is a subset of R2 consisting of vectors of the form To determine whether W is a sub space of R2. If W is a subspace, then give a geometric description of W. W = {x: x12 + x2 = 1}
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Chapter 3: Problem 11 Introduction to Linear Algebra 5
Problem 11E W is a subset of R3 consisting of vectors of the form Determine whether W is a subspace of R3 If W is a subspace, then give a geometric description of W. W = {x: x1 x2 = x3}
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Chapter 3: Problem 12 Introduction to Linear Algebra 5
Problem 12E W is a subset of R3 consisting of vectors of the form Determine whether W is a subspace of R3 If W is a subspace, then give a geometric description of W. W = {x: x1 = 2x3}
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Chapter 3: Problem 13 Introduction to Linear Algebra 5
Problem 13E W is a subset of R3 consisting of vectors of the form Determine whether W is a subspace of R3 If W is a subspace, then give a geometric description of W. W = {x: x12 = x1 + x2}
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Chapter 3: Problem 14 Introduction to Linear Algebra 5
Problem 14E W is a subset of R3 consisting of vectors of the form Determine whether W is a subspace of R3 If W is a subspace, then give a geometric description of W. W = {x: x2 = 0}
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Chapter 3: Problem 16 Introduction to Linear Algebra 5
Problem 16E W is a subset of R3 consisting of vectors of the form Determine whether W is a subspace of R3 If W is a subspace, then give a geometric description of W. W = {x: x3 = x2 = 2x1}
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Chapter 3: Problem 15 Introduction to Linear Algebra 5
Problem 15E W is a subset of R3 consisting of vectors of the form Determine whether W is a subspace of R3 If W is a subspace, then give a geometric description of W. W = {x: x1 = 2x3, x2 = -x3}
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Chapter 3: Problem 17 Introduction to Linear Algebra 5
Problem 17E W is a subset of R3 consisting of vectors of the form Determine whether W is a subspace of R3 If W is a subspace, then give a geometric description of W. W = {x: x2 = x3 = 0}
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Chapter 3: Problem 18 Introduction to Linear Algebra 5
Problem 18E Let abe a fixed vector in R3, and define W to be the subset of R3 given by w = [x: aTx = 0] Prove that W is a subspace of R3.
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Chapter 3: Problem 19 Introduction to Linear Algebra 5
Problem 19E Let abe a fixed vector in R3, and define W to be the subset of R3 given by w = [x: aTx = 0] Prove that W is a subspace of R3, where Give a geometric description for W.
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Chapter 3: Problem 20 Introduction to Linear Algebra 5
Problem 20E Let abe a fixed vector in R3, and define W to be the subset of R3 given by w = [x: aTx = 0] Prove that W is a subspace of R3, where Give a geometric description for W.
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Chapter 3: Problem 22 Introduction to Linear Algebra 5
Problem 22E Let a and b be fixed vectors in R3, and let W be the subset of R3 defined by w = {x: aT x = 0 and bT x = 0} Prove that W is a subspace of R3. For each choice of a and b, give a geometric description ofW.
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Chapter 3: Problem 21 Introduction to Linear Algebra 5
Problem 21E Let a and b be fixed vectors in R3, and let W be the subset of R3 defined by w = {x: aT x = 0 and bT x = 0} Prove that W is a subspace of R3.
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Chapter 3: Problem 24 Introduction to Linear Algebra 5
Problem 24E Let a and b be fixed vectors in R3, and let W be the subset of R3 defined by w = {x: aT x = 0 and bT x = 0} Prove that W is a subspace of R3. For each choice of a and b, give a geometric description ofW.
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Chapter 3: Problem 23 Introduction to Linear Algebra 5
Problem 23E Let a and b be fixed vectors in R3, and let W be the subset of R3 defined by w = {x: aT x = 0 and bT x = 0} Prove that W is a subspace of R3. For each choice of a and b, give a geometric description ofW.
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Chapter 3: Problem 25 Introduction to Linear Algebra 5
Problem 25E Let a and b be fixed vectors in R3, and let W be the subset of R3 defined by w = {x: aT x = 0 and bT x = 0} Prove that W is a subspace of R3. For each choice of a and b, give a geometric description ofW.
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Chapter 3: Problem 28 Introduction to Linear Algebra 5
Let In the statement of Theorem 1, replace each occurrence of with W. Illustrate with specific examples each of the ten properties of Theorem 1 that are not satisfied.
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Chapter 3: Problem 29 Introduction to Linear Algebra 5
Problem 29E In R3, a fine through the origin is the set of all points in R3 whose coordinates satisfy x1 = at, x2= bt, and x3 = ct, where t is a variable and a, b, and c are not all zero. Show that a line through the origin is a subspace of R3
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Chapter 3: Problem 30 Introduction to Linear Algebra 5
Problem 30E If U and V are subsets of Rn, then the set U + V is defined by U+ V = {x; x = u + v, u in U and v in V} Prove that if U and V are subspaces of Rn, then U + V is a subspace of Rn
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Chapter 3: Problem 31 Introduction to Linear Algebra 5
Problem 31E Let U and V be subspaces of Rn. Prove that the intersection, U ? V, is also a subspace of Rn
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Chapter 3: Problem 32 Introduction to Linear Algebra 5
Problem 32E Let U and V be the subspaces of R3 defined by U = {x: aT x = 0} and V = {x: bT x = 0} Where Demonstrate that the union, U ? V, is not a subspace of R3(see Let abe a fixed vector in R3, and define W to be the subset of R3 given by w = [x: aTx = 0] Prove that W is a subspace ofR3)
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Chapter 3: Problem 33 Introduction to Linear Algebra 5
Problem 33E Let U and V be subspaces of . a) Show that the union, U ? V, satisfies properties (s1) and (s3) of Theorem 2. b) If neither U nor V is a subset of the other, show that U ? V does not satisfy condition (s2) of Theorem 2. [Hint: Choose vectors u and v such that u is in U but not in V and v is in V but not in U. Assume that u + v is in either U or V and reach a contradiction.] (Reference Theorem 1)
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