Pure Mathematics For Beginners

Example 16.12:

1. Consider ℂ as a vector space over ℝ and define : ℂ → ℂ by ( ) = 5 , as we did in part 1 of
Example 16.1. The equation ( ) = 5 tells us that every nonzero vector is an eigenvector
corresponding to the eigenvalue = 5. ( ) = is equivalent to 5 = or ( − 5) = 0.
So, = 5 is the only eigenvalue. In particular, the standard basis vectors 1 and are
eigenvectors corresponding to the eigenvalue = 5. We have

(1) = 5 = 5 ⋅ 1 + 0 ⋅ .
( ) = 5 = 0 ⋅ 1 + 5 ⋅ .

So, as we saw in part 1 of Example 16.6, the matrix of with respect to the standard basis is
ℳ = [05 05], a diagonal matrix. Therefore, is diagonalizable.
2. Consider ℝ3 as a vector space over ℝ and define : ℝ3 → ℝ3 by

(( , , )) = (3 + , − 2 , 7 ).

Let’s find the eigenvalues and eigenvectors of .

We start by solving the equation (( , , )) = ( , , ). This equation is equivalent to the
three equations 3 + = , − 2 = , and 7 = . We work backwards. If ≠ 0, we get
= 7. If = 0 and ≠ 0, we get = 1. Finally, if = 0, = 0, and ≠ 0, we get = 3.

So, the eigenvalues of are 7, 1, and 3.

If we let = 7, we get 3 + = 7 , − 2 = 7 , and 7 = 7 . The equation − 2 = 7 is

equivalent to the equation 6 = – 2 , or = – 1 . The equation 3 + = 7 is equivalent to
3
1 1
4 = = – 3 , or = – 12 . So, if we let = – 12, we get the eigenvector 1 = (1, 4, – 12).

If we let = 1, we get 3 + = , − 2 = , and 7 = . The equation 7 = is equivalent

to the equation = 0. The equation − 2 = is then equivalent to = . The equation
3 + = is equivalent to 2 = – or = – 1 . So, if we let = – 2, we get the eigenvector

2

2 = (1, – 2, 0).

If we let = 3, we get 3 + = 3 , − 2 = 3 , and 7 = 3 . The equation 7 = 3 is
equivalent to the equation = 0. The equation − 2 = 3 is then equivalent to = 0. The
equation 3 + = 3 is then equivalent to = . So, if we let = 1, we get the eigenvector
3 = (1, 0, 0).

It follows that = {(1, 4, – 12), (1, – 2, 0), (1, 0, 0)} is a basis of eigenvectors of and we have

((1, 4, – 12)) = 7(1, 4, – 12)

((1, – 2, 0)) = 1(1, – 2, 0)

((1, 0, 0)) = 3(1, 0, 0)

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Therefore, the matrix of with respect to is ℳ ( ) = [0 1 0 ].

003

Since ℳ ( ) is a diagonal matrix, is diagonalizable.

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3. Recall from part 3 of Example 16.9, the linear transformation : ℝ2 → ℝ2 defined by
(( , )) = (– , ) (where ℝ2 is being viewed as a vector space over the field ℝ). We saw in
that example that this linear transformation has no eigenvalues. It follows that there is no basis
for ℝ2 such that the matrix of with respect to that basis is a diagonal matrix. In other words,
is not diagonalizable.

However, in the same example, we saw that the linear transformation : ℂ2 → ℂ2 defined by
: ℂ2 → ℂ2 by (( , )) = (– , ) (where ℂ2 is being viewed as a vector space over the field
ℂ) has eigenvalues and – with eigenvectors corresponding to these eigenvalues of (1, – )
and (1, ), respectively. So, we have

((1, – )) = (1, – )

((1, )) = – (1, )

So, the matrix of with respect to the basis = {(1, – ), (1, )} is ℳ ( ) = [0 –0 ], a
diagonal matrix. Therefore, is diagonalizable.

We finish with a Theorem that gives a sufficient condition for a linear transformation to be
diagonalizable.

Theorem 16.12: Let be an -dimensional vector space and let ∈ ℒ( ) have distinct eigenvalues.
Then is diagonalizable.

Proof: Suppose that dim = and ∈ ℒ( ) has the distinct eigenvalues 1, 2, … , , with

corresponding eigenvectors 1, 2, … , . By Theorem 16.11, 1, 2, … , are linearly independent. By

the note following Theorem 16.8, 1, 2, … , can be extended to a basis of . However, a basis of

has elements and therefore, { 1, 2, … , } is already a basis of . Since ( 1) = 1 1,
1 0 … 0

( 2) = 2 2,…, ( ) = , it follows that ℳ ( ) = [0⋮ 2 … 0⋮ ]. Since ℳ ( ) is a diagonal
⋮⋱

0 0 …

matrix, is diagonalizable. □

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Problem Set 16

Full solutions to these problems are available for free download here:

www.SATPrepGet800.com/PMFBXSG

LEVEL 1

1. Let and be vector spaces over ℝ. Determine if each of the following functions is a linear
transformation:
(i) : ℝ → ℝ defined by ( ) = 2 + 1
(ii) : ℝ → ℝ2 defined by ( ) = (2 , 3 )

(iii) ℎ: ℝ3 → ℝ3 defined by ℎ(( , , )) = ( + , + , − )

2. Compute each of the following:

(i) [20 0 – 43] ⋅ 1 1 30
1 [1 –4 2 0]
2 1 –4
0

–4
(ii) [3 – 1 5] ⋅ [– 7]

2

–4
(iii) [– 7] ⋅ [3 – 1 5]

2

1 0 1
(iv) [ ] ⋅ [0 2 0].

ℎ 3 1 4

LEVEL 2

3. Consider ℂ as a vector space over itself. Give an example of a function : ℂ → ℂ such that is
additive, but not a linear transformation. Then give an example of vector spaces and and a
homogenous function : → that is not a linear transformation.

LEVEL 3

4. Let = { 2 + + | , , ∈ ℝ} be the vector space of polynomials of degree 2 with real
coefficients (see part 3 of Example 8.3 from Lesson 8). Define the linear transformation
: → by ( 2 + + ) = 2 + . Find the matrix of with respect to each of the
following bases:

(i) The standard basis = {1, , 2}

(ii) = { + 1, 2 + 1, 2 + }

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5. Let and be vector spaces with finite-dimensional, let ≤ , and let ∈ ℒ( , ). Prove
that there is an ∈ ℒ( , ) such that ( ) = ( ) for all ∈ .

LEVEL 4

6. Let : → be a linear transformation and let 1, 2, … , ∈ . Prove the following:
(i) If is injective and 1, 2, … , are linearly independent in , then
( 1), ( 2), … , ( ) are linearly independent in .
(ii) If is surjective and span{ 1, 2, … , } = , then span{ ( 1), ( 2), … , ( )} = .

7. Determine if each linear transformation is diagonalizable:
(i) : ℝ2 → ℝ2 defined by (( , )) = ( , 2 )
(ii) : ℂ2 → ℂ2 defined by (( , )) = ( + , − ).

8. Let and be vector spaces over a field . Prove that ℒ( , ) is a vector space over , where
addition and scalar multiplication are defined as in Theorem 16.2.

9. Let be a vector space over a field . Prove that ℒ( ) is a linear algebra over , where addition
and scalar multiplication are defined as in Theorem 16.2 and vector multiplication is given by
composition of linear transformations.

10. Let : → and : → be linear transformations such that = and = . Prove
that and are bijections and that = −1.

11. Let and be finite-dimensional vector spaces and let ∈ ℒ( , ). Prove the following:
(i) If dim < dim , then is not surjective
(ii) If dim > dim , then is not injective.

12. Prove that two finite-dimensional vector spaces over a field are isomorphic if and only if they
have the same dimension.

13. Let ∈ ℒ( ) be invertible and let ∈ ∖ {0}. Prove that is an eigenvalue of if and only if
1 is an eigenvalue of −1.



LEVEL 5

14. Let be a vector space with dim > 1. Show that { ∈ ℒ( ) | is not invertible} ≰ ℒ( ).

15. Let be an -dimensional vector space over a field . Prove that there is a linear algebra
isomorphism : ℒ( ) → .

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Abelian, 33 INDEX Conjunctive elimination, 114
Abelian group, 35 Conjunctive introduction, 114
Absolute value, 82 Canonical representation, 162 Connective, 9
Absorption law, 109 Cantor-Schroeder-Bernstein Constant, 137
Accessible space, 197 Theorem, 133 Constant function, 125
Accumulation point, 91 Cantor’s Theorem, 131 Constructive dilemma, 114
Additive function, 234 Cardinality, 20 Continuity, 174, 177, 204, 226
Algebraically closed field, 78 Cartesian product, 30, 119 Continuous at a point, 174,
Almost ring, 146 Chains of topologies, 191 177, 205, 226
Angle, 212 Circle, 85, 212 Continuous function, 174, 204
Angle in standard position, 212 Circumference, 212 Contradiction, 109
Antireflexive, 120 Clopen, 200 Contrapositive, 113
Antisymmetric, 120 Closed disk, 86 Contrapositive law, 109
Archimedean Property, 60 Closed downwards, 33, 98 Converse, 113
Argument of a complex Closed interval, 64 Corollary, 130
number, 216 Closed set, 74, 89, 189 Cosine, 214
Associative, 32 Closing statement, 21 Countable, 131
Associative law, 109 Closure, 31, 35 Counterexample, 31
Assumption, 109 Coarser topology, 190 Cover of a topology, 194
Atomic statement, 9, 107 Codomain, 125 Covering, 203
Automorphism, 146 Cofinite topology, 198 Cycle diagram, 148
Axiom of Extensionality, 26 Common divisor, 159 Cycle notation, 148
Ball, 201 Common factor, 159 De Moivre’s Theorem, 217
Base case, 44 Common multiple, 159 De Morgan’s laws, 11, 77, 109
Basis, 102, 104 Commutative, 33 Deleted neighborhood, 87
Basis for a topology, 192 Commutative group, 35 Dense, 61
Biconditional, 11 Commutative law, 109 Density Theorem, 61
Biconditional elimination, 114 Compact space, 203 Denumerable, 131
Biconditional introduction, 114 Comparability condition, 124 Dependence relation, 104
Biconditional law, 109 Complement, 74, 89 Derivation, 114
Bijection, 126 Complete prime factorization, Destructive dilemma, 114
Bijective function, 126 164 Diagonal entry, 250
Binary connective, 10 Completeness, 58 Diagonal matrix, 250
Binary operation, 30 Completeness Property, 60 Diagonalizable, 250
Binary relation, 119, 137 Complex number, 78 Difference identity, 232
Boundary point, 92 Composite number, 152 Dilation, 234
Bounded, 58 Composite function, 129 Dimension, 244
Bounded above, 58 Compound statement, 9 Discrete topology, 190
Bounded below, 58 Conclusion, 109, 111 Disjoint, 25, 70
Bounded interval, 64 Conditional, 11 Disjunction, 11
Canonical form, 162 Conditional law, 109 Disjunctive introduction, 114
Conjugate, 81, 147
Conjunction, 11

255

Disjunctive resolution, 114 Finite-dimensional vector Infinite set, 19
Disjunctive syllogism, 114 space, 244 Initial ray, 212
Disk, 86 Finite sequence, 126 Injection, 126
Distance, 83 Fixed point, 221 Injective function, 126
Distance function, 201 Function, 124, 218 Integer, 19
Distributive, 39, 77 Fundamental Theorem of Interior point, 92
Distributive law, 107 Arithmetic, 152 Intersection, 24, 66, 69
Distributivity, 40 Gaussian integer, 150 Intersection containment
Divides, 42, 152 GCD, 159 property, 194
Divisible, 42, 152 Greatest common divisor, 159 Interval, 64
Divisibility, 41 Greatest common factor, 159 Invalid, 112
Division Algorithm, 155, 156 Greatest lower bound, 58 Invariant, 220
Divisor, 42, 152 Group, 34 Invariant subspace, 247
Domain, 125, 137 Group homomorphism, 142 Inverse, 34, 113
Double negation, 108 Half-open interval, 64 Inverse function, 127
Eigenvalue, 247 Hausdorff space, 198 Invertible, 34
Eigenvector, 247 Homeomorphic spaces, 207 Isomorphism, 55, 145
Element, 19 Homeomorphism, 207 Kernel, 146, 244
Empty set, 20 Homogenous function, 234 Kolmogorov space, 197
Equinumerosity, 130 Homomorphism, 142 LCM, 159
Equivalence class, 122 Horizontal strip, 169 Least common multiple, 159
Equivalence relation, 121 Hypothesis, 109 Least upper bound, 58
Euclidean Algorithm, 165 Hypothetical syllogism, 114 Left distributivity, 39
Euler’s formula, 216 Ideal, 149 Lemma, 250
Even, 41, 47 Identity, 34 Limit, 172, 176, 177, 223
Exclusive or, 17 Identity function, 130 Limits involving infinity, 183
Exponential form of a complex Identity law, 109 Linear algebra, 237
number, 216 Image, 146, 204, 244 Linear combination, 101, 103,
Extended Complex Plane, 229 Imaginary part, 79 160
Factor, 42, 152 Implication, 11 Linear dependence, 102, 104
Factor tree, 162 Inclusion map, 134 Linear equation, 78
Factorial, 154 Incomparable topologies, 190 Linear function, 179
Factorization, 153 Indiscrete topology, 190 Linear independence, 102, 104
Fallacy, 112 Induced topology, 202 Linear transformation, 234
Fallacy of the converse, 113 Induction, 43 Linearly ordered set, 124
Fallacy of the inverse, 113 Inductive hypothesis, 44 Logical argument, 111
Fence-post formula, 20 Inductive step, 44 Logical connective, 9
Field, 41, 50 Infimum, 58, 59 Logical equivalence, 108
Field axioms, 50, 51 Infinite closed interval, 64 Lower bound, 58
Field homomorphism, 143 Infinite interval, 64 Matrix, 97, 239
Finer topology, 190 Infinite limit, 183 Matrix addition, 97
Finitary operation, 137 Infinite open interval, 64 Matrix of a linear
Finitary relation, 137 Infinite sequence, 126 transformation, 242

256

Matrix multiplication, 240, 241 Pairwise disjoint, 70, 121 Redundancy law, 109
Matrix scalar multiplication, 97 Pairwise relatively prime, 160 Reflection, 221
Metric, 201 Parity, 121 Reflexive, 29, 120
Metric space, 201 Partial binary operation, 31 Regular space, 200
Metrizable space, 202 Partial ordering, 124 Relation, 119, 120, 137
Modulus, 82 Partially ordered set, 124 Relatively prime, 159
Modus ponens, 112, 114 Partition, 121 Representative of equivalence
Modus tollens, 114 Permutation, 148 class, 123
Monoid, 34 Point at infinity, 228 Riemann sphere, 228
Monoid homomorphism, 142 Polar form of a complex Right distributivity, 39
Monotonic function, 143 number, 216 Ring, 39
Multiple, 42, 152 Polynomial equation, 78 Ring axioms, 40
Mutually exclusive, 25 Polynomial ring, 151 Ring homomorphism, 143
Mutually relatively prime, 160 Poset, 124 Ring ideal, 149
Natural number, 19 Positive square root, 82 Ring with identity, 40
Negation, 11 Power set, 23 Rng, 147
Negation law, 109 Premise, 109, 111 Root of a complex number, 218
Negative identities, 215 Prime factorization, 153 Roots of unity, 218
Neighborhood, 86 Prime number, 152 Rotation, 222
Nondiagonal entry, 250 Principle of Mathematical Rule of inference, 112
Normal, 147 Induction, 43 SACT, 45
Normal space, 200 Principle root, 218 Scalar multiplication, 93, 95
Normal subgroup, 147 Product, 41 Semigroup, 32
North pole, 228 Product topology, 210 Semigroup homomorphism,
Null space, 244 Proof, 111 142
Nullity, 246 Proof by contradiction, 44 Semiring, 41
Odd, 47 Proof by contrapositive, 129 Separation axioms, 197
One-sided limit, 185 Proposition, 9, 107 Sequence, 126
One-to-one function, 126 Propositional variable, 10, 107 Set, 19
Onto, 126 Punctured disk, 87 Set-builder notation, 20
Open ball, 201 Pure imaginary number, 79 Set complement, 74
Open covering, 203 Pythagorean identity, 215 Set difference, 66
Open disk, 86 Pythagorean Theorem, 56 Sigma notation, 241
Open interval, 64 Quadrantal angles, 214 Simple subspace, 247
Open rectangle, 170 Quadratic equation, 78 Sine, 214
Open set, 71, 87, 189 Quotient, 35, 81, 155 Soundness, 113
Opening statement, 21 Radian measure, 212 South pole, 228
Order homomorphism, 143 Range, 125, 244 Span, 101, 103
Ordered field, 52 Rank, 246 Square matrix, 240
Ordered pair, 118 Rational number, 35 Square root, 82, 218
Ordered ring, 52 Ray, 212 Standard Advanced Calculus
Ordered tuple, 118 Real number, 60, 79 Trick, 45
Ordering, 124 Real part, 79

257

Standard form of a complex Supremum, 59 Trivial topology, 190
number, 78, 216 Surjection, 126 Truth table, 12
Standard topology, 192 Surjective function, 126 Type, 138
Statement, 9, 107 Surjectively invariant, 220 Unary connective, 10
Strict linearly ordered set, 124 Symmetric, 29, 120 Unary relation, 120, 137
Strict partial ordering, 124 Symmetric difference, 66 Uncountable, 131
Strict partially ordered set, 124 Tangent, 214 Uniform continuity, 180, 227
Strict poset, 124 Tautologically implies, 112 Uniformly continuous, 180, 227
Strip, 169 Tautology, 22, 109 Union, 24, 66, 69
Strong Induction, 49 Terminal ray, 212 Unit circle, 212
Subbasis for a topology, 196 Ternary relation, 120, 137 Unital ring, 40
Subfield, 80, 141 Theorem, 21 Universal Quantifier, 21
Subgroup, 140 Tichonov space, 197 Universal set, 21
Submonoid, 139 Topological equivalence, 207 Universal statement, 33, 98
Subring, 140 Topological invariant, 209 Unordered pair, 118
Subsemigroup, 139 Topological property, 209 Upper bound, 58
Subset, 20 Topological space, 189 Valid, 112
Subspace, 98 Topology, 85, 189 Vector, 79, 95
Subspace topology, 210 Totally ordered set, 124 Vector multiplication, 237
Substatement, 107 Transitive, 24, 120 Vector space, 93
Substitution of logical Transitivity of logical Venn diagram, 21
equivalents, 109 equivalence, 109 Vertical strip, 169
Substitution of sentences, 109 Translation, 220 Weight, 101, 103, 160
Substructure, 139 Tree diagram, 23 Well-defined, 123
Sum, 41 Triangle Inequality, 84 Well Ordering Principle, 43
Sum identities, 215 Trichotomy, 124 Without loss of generality, 73
Summation, 241 Trigonometric functions, 214 Wrapping function, 214

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About the Author

Dr. Steve Warner, a New York native, earned his Ph.D. at Rutgers University in Pure Mathematics in
May 2001. While a graduate student, Dr. Warner won the TA Teaching
Excellence Award.
After Rutgers, Dr. Warner joined the Penn State Mathematics
Department as an Assistant Professor and in September 2002, he
returned to New York to accept an Assistant Professor position at Hofstra
University. By September 2007, Dr. Warner had received tenure and was
promoted to Associate Professor. He has taught undergraduate and
graduate courses in Precalculus, Calculus, Linear Algebra, Differential
Equations, Mathematical Logic, Set Theory, and Abstract Algebra.
From 2003 – 2008, Dr. Warner participated in a five-year NSF grant, “The

MSTP Project,” to study and improve mathematics and science curriculum in poorly performing junior
high schools. He also published several articles in scholarly journals, specifically on Mathematical Logic.
Dr. Warner has nearly two decades of experience in general math tutoring and tutoring for
standardized tests such as the SAT, ACT, GRE, GMAT, and AP Calculus exams. He has tutored students
both individually and in group settings.
In February 2010 Dr. Warner released his first SAT prep book “The 32 Most Effective SAT Math
Strategies,” and in 2012 founded Get 800 Test Prep. Since then Dr. Warner has written books for the
SAT, ACT, SAT Math Subject Tests, AP Calculus exams, and GRE.
Dr. Steve Warner can be reached at

[email protected]

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