Linear Algebra

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280 Chapter Three. Maps Between Spaces0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,p,0,1,0; # 4-30,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,q,0,0,1]; # 3-4w = A * v;endfunctionThen the Octave session was this.> v0=[1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0]> p=.5> v1=markov(p,v0)> v2=markov(p,v1)...Translating to another computer algebra system should be easy — all have commandssimilar to these.
Topic: Orthonormal Matrices 281Topic: Orthonormal MatricesIn The Elements, Euclid considers two figures to be the same if they have thesame size and shape. That is, the triangles below are not equal because theyare not the same set of points. But they are congruent — essentially indistinguishablefor Euclid’s purposes — because we can imagine picking the plane up,sliding it over and rotating it a bit, although not warping or stretching it, andthen putting it back down, to superimpose the first figure on the second. (Euclidnever explicitly states this principle but he uses it often [Casey].)P 1P 2P 3Q 1Q 2Q 3In modern terminology, “picking the plane up . . . ” means considering a mapfrom the plane to itself. Euclid has limited consideration to only certain transformationsof the plane, ones that may possibly slide or turn the plane but notbend or stretch it. Accordingly, we define a map f : R 2 → R 2 to be distancepreservingor a rigid motion or an isometry, if for all points P 1 , P 2 ∈ R 2 , thedistance from f(P 1 ) to f(P 2 ) equals the distance from P 1 to P 2 . We also definea plane figure to be a set of points in the plane and we say that two figuresare congruent if there is a distance-preserving map from the plane to itself thatcarries one figure onto the other.Many statements from Euclidean geometry follow easily from these definitions.Some are: (i) collinearity is invariant under any distance-preserving map(that is, if P 1 , P 2 , and P 3 are collinear then so are f(P 1 ), f(P 2 ), and f(P 3 )),(ii) betweeness is invariant under any distance-preserving map (if P 2 is betweenP 1 and P 3 then so is f(P 2 ) between f(P 1 ) and f(P 3 )), (iii) the property ofbeing a triangle is invariant under any distance-preserving map (if a figure is atriangle then the image of that figure is also a triangle), (iv) and the property ofbeing a circle is invariant under any distance-preserving map. In 1872, F. Kleinsuggested that Euclidean geometry can be characterized as the study of propertiesthat are invariant under these maps. (This forms part of Klein’s ErlangerProgram, which proposes the organizing principle that each kind of geometry —Euclidean, projective, etc. — can be described as the study of the propertiesthat are invariant under some group of transformations. The word ‘group’ heremeans more than just ‘collection’, but that lies outside of our scope.)We can use linear algebra to characterize the distance-preserving maps ofthe plane.First, there are distance-preserving transformations of the plane that are notlinear. The obvious example is this translation.( xy)↦→( ( x 1+ =y)0)( ) x + 1y
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Linear AlgebraJim Hefferon( 13)( 21
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PrefaceThis book helps students to
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the material, but it’s only that
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Chapter Three: Maps Between Spaces
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Chapter OneLinear SystemsISolving L
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Section I. Solving Linear Systems 3
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Topic: Computer Algebra Systems 631
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Topic: Input-Output Analysis 65For
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Topic: Input-Output Analysis 67(c)
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Topic: Accuracy of Computations 69(
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Topic: Analyzing Networks 73differe
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Topic: Analyzing Networks 75Kirchof
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80 Chapter Two. Vector SpacesIDefin
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84 Chapter Two. Vector SpacesWarnin
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88 Chapter Two. Vector SpacesExerci
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98 Chapter Two. Vector Spaceš 2.2
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128 Chapter Two. Vector SpacesSo ou
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134 Chapter Two. Vector SpacesProof
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138 Chapter Two. Vector Spaces(b) A
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140 Chapter Two. Vector SpacesTopic
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144 Chapter Two. Vector Spaces(To s
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334 Chapter Four. DeterminantsFor a
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336 Chapter Four. Determinantsvecto
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Chapter FiveSimilarityWhile studyin
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Section I. Complex Vector Spaces 34
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Section II. Similarity 347IISimilar
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Section II. Similarity 3491.7 Consi
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Section II. Similarity 351Proof. Th
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Section II. Similarity 353(b) Is th
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Section II. Similarity 355The next
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Section II. Similarity 3573.13 Exam
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Section II. Similarity 3593.23 Find
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Section III. Nilpotence 361IIINilpo
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Section III. Nilpotence 363and this
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Section III. Nilpotence 365Proof. W
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Section III. Nilpotence 367The new
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Section III. Nilpotence 369Now, wit
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Section IV. Jordan Form 375IVJordan
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Section IV. Jordan Form 381̌ 1.17
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Section IV. Jordan Form 3832.3 Exam
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Section IV. Jordan Form 385their fi
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Section IV. Jordan Form 387Proof. S
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Topic: Method of Powers 395Topic: M
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Topic: Method of Powers 397(a)( )1
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Topic: Linear Recurrences 401Topic:
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Topic: Linear Recurrences 403And, i
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Topic: Linear Recurrences 4076 (Thi
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AppendixMathematics is made of argu
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A-3shows that Q holds whenever P do
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A-5more factors than any other’ w
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A-7IV.6 Sets, Functions, and Relati
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A-9We sometimes instead use the not
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A-11S 1 S 2S 0. . .S 3Thus, the fir
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Bibliography[Abbot] Edwin Abbott, F
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[Feller] William Feller, An Introdu
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[Programmer’s Ref.] Microsoft Pro
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composition, A-9self, 361computer a
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Kirchhoff’s Laws, 73Klein, F., 28
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at infinity, 337in projective plane
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vacuously true, A-2value, A-8Vander
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Linear Algebra

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