Answers to Exercises
Linear Algebra
Jim Hefferon
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,Contents
Chapter One: Linear Systems 4
Subsection One.I.1: Gauss’ Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Subsection One.I.2: Describing the Solution Set ............................................................................. 10
Subsection One.I.3: General = Particular + Homogeneous ......................................................... 14
Subsection One.II.1: Vectors in Space.............................................................................................. 17
Subsection One.II.2: Length and Angle Measures .......................................................................... 20
Subsection One.III.1: Gauss-Jordan Reduction ............................................................................... 25
Subsection One.III.2: Row Equivalence ........................................................................................... 27
Topic: Computer Algebra Systems ................................................................................................... 31
Topic: Input-Output Analysis ........................................................................................................... 33
Topic: Accuracy of Computations ..................................................................................................... 33
Topic: Analyzing Networks ............................................................................................................... 34
Chapter Two: Vector Spaces 36
Subsection Two.I.1: Definition and Examples ................................................................................. 37
Subsection Two.I.2: Subspaces and Spanning Sets ......................................................................... 40
Subsection Two.II.1: Definition and Examples................................................................................ 46
Subsection Two.III.1: Basis ............................................................................................................... 53
Subsection Two.III.2: Dimension ...................................................................................................... 58
Subsection Two.III.3: Vector Spaces and Linear Systems ............................................................. 61
Subsection Two.III.4: Combining Subspaces................................................................................... 66
Topic: Fields ........................................................................................................................................ 69
Topic: Crystals..................................................................................................................................... 70
Topic: Dimensional Analysis .............................................................................................................. 71
Chapter Three: Maps Between Spaces 73
Subsection Three.I.1: Definition and Examples............................................................................... 75
Subsection Three.I.2: Dimension Characterizes Isomorphism ...................................................... 83
Subsection Three.II.1: Definition ...................................................................................................... 85
Subsection Three.II.2: Rangespace and Nullspace .......................................................................... 90
Subsection Three.III.1: Representing Linear Maps with Matrices ................................................ 95
Subsection Three.III.2: Any Matrix Represents a Linear Map.................................................... 103
Subsection Three.IV.1: Sums and Scalar Products ....................................................................... 107
Subsection Three.IV.2: Matrix Multiplication ............................................................................... 108
Subsection Three.IV.3: Mechanics of Matrix Multiplication ........................................................ 112
Subsection Three.IV.4: Inverses ..................................................................................................... 116
Subsection Three.V.1: Changing Representations of Vectors ...................................................... 121
Subsection Three.V.2: Changing Map Representations ............................................................... 124
Subsection Three.VI.1: Orthogonal Projection Into a Line ......................................................... 128
Subsection Three.VI.2: Gram-Schmidt Orthogonalization ........................................................... 131
Subsection Three.VI.3: Projection Into a Subspace ..................................................................... 137
Topic: Line of Best Fit...................................................................................................................... 143
Topic: Geometry of Linear Maps .................................................................................................... 147
Topic: Markov Chains....................................................................................................................... 150
Topic: Orthonormal Matrices .......................................................................................................... 157
Chapter Four: Determinants 158
Subsection Four.I.1: Exploration ..................................................................................................... 159
Subsection Four.I.2: Properties of Determinants .......................................................................... 161
Subsection Four.I.3: The Permutation Expansion ........................................................................ 164
Subsection Four.I.4: Determinants Exist ........................................................................................ 166
Subsection Four.II.1: Determinants as Size Functions ................................................................. 168
Subsection Four.III.1: Laplace’s Expansion ................................................................................... 171
, Topic: Cramer’s Rule........................................................................................................................ 174
4 Linear Algebra, by Hefferon
Topic: Speed of Calculating Determinants .................................................................................... 175
Topic: Projective Geometry ............................................................................................................. 176
Chapter Five: Similarity 178
Subsection Five.II.1: Definition and Examples .............................................................................. 179
Subsection Five.II.2: Diagonalizability............................................................................................ 182
Subsection Five.II.3: Eigenvalues and Eigenvectors ..................................................................... 186
Subsection Five.III.1: Self-Composition ......................................................................................... 190
Subsection Five.III.2: Strings .......................................................................................................... 192
Subsection Five.IV.1: Polynomials of Maps and Matrices ........................................................... 196
Subsection Five.IV.2: Jordan Canonical Form .............................................................................. 203
Topic: Method of Powers ................................................................................................................ 210
Topic: Stable Populations ................................................................................................................ 210
Topic: Linear Recurrences ............................................................................................................... 210
, Chapter One: Linear Systems
Subsection One.I.1: Gauss’ Method
One.I.1.16 Gauss’ method can be performed in different ways, so these simply exhibit one possible
way to get the answer.
(a) Gauss’ method
−(1/2)ρ1+ρ2 2x + 3y = 7
—→
— (5/2)y = —15/2
gives that the solution is y = 3 and x = 2.
(b) Gauss’ method here
x — z=0 x — z=0
−3ρ1+ρ2 −ρ2+ρ3
—→ y + 3z = 1 —→ y + 3z = 1
ρ1+ρ3
y =4 —3z = 3
gives x = —1, y = 4, and z = —1.
One.I.1.17 (a) Gaussian reduction
−(1/2)ρ1+ρ2 2x + 2y = 5
—→
—5y = —5/2
shows that y = 1/2 and x = 2 is the unique solution.
(b) Gauss’ method
ρ1+ρ2 —x + y = 1
—→
2y = 3
gives y = 3/2 and x = 1/2 as the only solution.
(c) Row reduction
−ρ1+ρ2 x — 3y + z = 1
—→
4y + z = 13
shows, because the variable z is not a leading variable in any row, that there are many solutions.
(d) Row reduction
−3ρ1+ρ2 —x — y = 1
—→
0 = —1
shows that there is no solution.
(e) Gauss’ method
x + y — z = 10 x+ y — z = 10 x+ y— z = 10
ρ1↔ρ4 2x — 2y + z = 0 −2ρ1+ρ2 —4y + 3z = —20 −(1/4)ρ2+ρ3 —4y + 3z = —20
—→ —→ —→
x +z= 5 −ρ1+ρ3 —y + 2z = —5 ρ2+ρ4 (5/4)z = 0
4y + z = 20 4y + z = 20 4z = 0
gives the unique solution (x, y, z) = (5, 5, 0).
(f) Here Gauss’ method gives
2x + z+ w= 5 2x + z+ w= 5
−(3/2)ρ1+ρ3 y — w= —1 −ρ2+ρ4 y — w= —1
—→ —→
−2ρ 1 +ρ 4 — (5/2)z — (5/2)w = —15/2 — (5/2)z — (5 /2)w = —15 /2
y — w= —1 0= 0
which shows that there are many solutions.
One.I.1.18 (a) From x = 1 — 3y we get that 2(1 — 3y) + y = —3, giving y = 1.
(b) From x = 1 — 3y we get that 2(1 — 3y) + 2y = 0, leading to the conclusion that y = 1/2.
Users of this method must check any potential solutions by substituting back into all the equations.
Linear Algebra
Jim Hefferon
¡1¢
3
¡2¢
1
1 2
¯3 1¯
¡1¢
x1 ·
3
¡2¢
1
¯x · 1 2¯
¯x · 3 1¯
¯6 2¯
¯8 1¯
,Contents
Chapter One: Linear Systems 4
Subsection One.I.1: Gauss’ Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Subsection One.I.2: Describing the Solution Set ............................................................................. 10
Subsection One.I.3: General = Particular + Homogeneous ......................................................... 14
Subsection One.II.1: Vectors in Space.............................................................................................. 17
Subsection One.II.2: Length and Angle Measures .......................................................................... 20
Subsection One.III.1: Gauss-Jordan Reduction ............................................................................... 25
Subsection One.III.2: Row Equivalence ........................................................................................... 27
Topic: Computer Algebra Systems ................................................................................................... 31
Topic: Input-Output Analysis ........................................................................................................... 33
Topic: Accuracy of Computations ..................................................................................................... 33
Topic: Analyzing Networks ............................................................................................................... 34
Chapter Two: Vector Spaces 36
Subsection Two.I.1: Definition and Examples ................................................................................. 37
Subsection Two.I.2: Subspaces and Spanning Sets ......................................................................... 40
Subsection Two.II.1: Definition and Examples................................................................................ 46
Subsection Two.III.1: Basis ............................................................................................................... 53
Subsection Two.III.2: Dimension ...................................................................................................... 58
Subsection Two.III.3: Vector Spaces and Linear Systems ............................................................. 61
Subsection Two.III.4: Combining Subspaces................................................................................... 66
Topic: Fields ........................................................................................................................................ 69
Topic: Crystals..................................................................................................................................... 70
Topic: Dimensional Analysis .............................................................................................................. 71
Chapter Three: Maps Between Spaces 73
Subsection Three.I.1: Definition and Examples............................................................................... 75
Subsection Three.I.2: Dimension Characterizes Isomorphism ...................................................... 83
Subsection Three.II.1: Definition ...................................................................................................... 85
Subsection Three.II.2: Rangespace and Nullspace .......................................................................... 90
Subsection Three.III.1: Representing Linear Maps with Matrices ................................................ 95
Subsection Three.III.2: Any Matrix Represents a Linear Map.................................................... 103
Subsection Three.IV.1: Sums and Scalar Products ....................................................................... 107
Subsection Three.IV.2: Matrix Multiplication ............................................................................... 108
Subsection Three.IV.3: Mechanics of Matrix Multiplication ........................................................ 112
Subsection Three.IV.4: Inverses ..................................................................................................... 116
Subsection Three.V.1: Changing Representations of Vectors ...................................................... 121
Subsection Three.V.2: Changing Map Representations ............................................................... 124
Subsection Three.VI.1: Orthogonal Projection Into a Line ......................................................... 128
Subsection Three.VI.2: Gram-Schmidt Orthogonalization ........................................................... 131
Subsection Three.VI.3: Projection Into a Subspace ..................................................................... 137
Topic: Line of Best Fit...................................................................................................................... 143
Topic: Geometry of Linear Maps .................................................................................................... 147
Topic: Markov Chains....................................................................................................................... 150
Topic: Orthonormal Matrices .......................................................................................................... 157
Chapter Four: Determinants 158
Subsection Four.I.1: Exploration ..................................................................................................... 159
Subsection Four.I.2: Properties of Determinants .......................................................................... 161
Subsection Four.I.3: The Permutation Expansion ........................................................................ 164
Subsection Four.I.4: Determinants Exist ........................................................................................ 166
Subsection Four.II.1: Determinants as Size Functions ................................................................. 168
Subsection Four.III.1: Laplace’s Expansion ................................................................................... 171
, Topic: Cramer’s Rule........................................................................................................................ 174
4 Linear Algebra, by Hefferon
Topic: Speed of Calculating Determinants .................................................................................... 175
Topic: Projective Geometry ............................................................................................................. 176
Chapter Five: Similarity 178
Subsection Five.II.1: Definition and Examples .............................................................................. 179
Subsection Five.II.2: Diagonalizability............................................................................................ 182
Subsection Five.II.3: Eigenvalues and Eigenvectors ..................................................................... 186
Subsection Five.III.1: Self-Composition ......................................................................................... 190
Subsection Five.III.2: Strings .......................................................................................................... 192
Subsection Five.IV.1: Polynomials of Maps and Matrices ........................................................... 196
Subsection Five.IV.2: Jordan Canonical Form .............................................................................. 203
Topic: Method of Powers ................................................................................................................ 210
Topic: Stable Populations ................................................................................................................ 210
Topic: Linear Recurrences ............................................................................................................... 210
, Chapter One: Linear Systems
Subsection One.I.1: Gauss’ Method
One.I.1.16 Gauss’ method can be performed in different ways, so these simply exhibit one possible
way to get the answer.
(a) Gauss’ method
−(1/2)ρ1+ρ2 2x + 3y = 7
—→
— (5/2)y = —15/2
gives that the solution is y = 3 and x = 2.
(b) Gauss’ method here
x — z=0 x — z=0
−3ρ1+ρ2 −ρ2+ρ3
—→ y + 3z = 1 —→ y + 3z = 1
ρ1+ρ3
y =4 —3z = 3
gives x = —1, y = 4, and z = —1.
One.I.1.17 (a) Gaussian reduction
−(1/2)ρ1+ρ2 2x + 2y = 5
—→
—5y = —5/2
shows that y = 1/2 and x = 2 is the unique solution.
(b) Gauss’ method
ρ1+ρ2 —x + y = 1
—→
2y = 3
gives y = 3/2 and x = 1/2 as the only solution.
(c) Row reduction
−ρ1+ρ2 x — 3y + z = 1
—→
4y + z = 13
shows, because the variable z is not a leading variable in any row, that there are many solutions.
(d) Row reduction
−3ρ1+ρ2 —x — y = 1
—→
0 = —1
shows that there is no solution.
(e) Gauss’ method
x + y — z = 10 x+ y — z = 10 x+ y— z = 10
ρ1↔ρ4 2x — 2y + z = 0 −2ρ1+ρ2 —4y + 3z = —20 −(1/4)ρ2+ρ3 —4y + 3z = —20
—→ —→ —→
x +z= 5 −ρ1+ρ3 —y + 2z = —5 ρ2+ρ4 (5/4)z = 0
4y + z = 20 4y + z = 20 4z = 0
gives the unique solution (x, y, z) = (5, 5, 0).
(f) Here Gauss’ method gives
2x + z+ w= 5 2x + z+ w= 5
−(3/2)ρ1+ρ3 y — w= —1 −ρ2+ρ4 y — w= —1
—→ —→
−2ρ 1 +ρ 4 — (5/2)z — (5/2)w = —15/2 — (5/2)z — (5 /2)w = —15 /2
y — w= —1 0= 0
which shows that there are many solutions.
One.I.1.18 (a) From x = 1 — 3y we get that 2(1 — 3y) + y = —3, giving y = 1.
(b) From x = 1 — 3y we get that 2(1 — 3y) + 2y = 0, leading to the conclusion that y = 1/2.
Users of this method must check any potential solutions by substituting back into all the equations.
