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Linear Algebra Jim Hefferon ¡ ¢ 1 3 ¡ ¢ 2 1 ¯ ¯ ¯ ¯ ¯1 2¯ ¯3 1¯ ¡ ¢ 1 x ¢ 1 3 ¡ ¢ 2 1 ¯ ¯ ¯ ¯ ¯x ¢ 1 2¯ ¯ ¯ x ¢ 3 1 ¡ ¢ 6 8 ¡ ¢ 2 1 ¯ ¯ ¯ ¯ ¯6 2¯ ¯8 1¯ Notation R real numbers N natural numbers: f0; 1; 2;:::g ¯ C complex numbers f::: ¯ :::g set of . such that . h:::i sequence; like a set but order matters V; W; U vector spaces ~v; ~w vectors ~0, ~0V zero vector, zero vector of V B; D bases n En = h~e1; : : : ; ~eni standard basis for R ¯;~ ~± basis vectors RepB(~v) matrix representing the vector Pn set of n-th degree polynomials Mn£m set of n£m matrices [S] span of the set S M © N direct sum of subspaces V =» W isomorphic spaces h; g homomorphisms, linear maps H; G matrices t; s transformations; maps from a space to itself T;S square matrices RepB;D(h) matrix representing the map h hi;j matrix entry from row i, column j jT j determinant of the matrix T R(h); N (h) rangespace and nullspace of the map h R1(h); N1(h) generalized rangespace and nullspace Lower case Greek alphabet name character name character name character alpha ® iota ¶ rho ½ beta ¯ kappa · sigma σ gamma γ lambda ¸ tau ¿ delta ± mu ¹ upsilon υ epsilon ² nu º phi Á zeta ³ xi » chi  eta ´ omicron o psi à theta θ pi ¼ omega ! Cover. This is Cramer’s Rule for the system x + 2y = 6, 3x + y = 8. The size of the first box is the determinant shown (the absolute value of the size is the area). The size of the second box is x times that, and equals the size of the final box. Hence, x is the final determinant divided by the first determinant. Contents Chapter One: Linear Systems 1 I Solving Linear Systems ........................ 1 1 Gauss’ Method ........................... 2 2 Describing the Solution Set .................... 11 3 General = Particular + Homogeneous .............. 20 II Linear Geometry of n-Space ..................... 32 1 Vectors in Space .......................... 32 2 Length and Angle Measures¤ ................... 38 III Reduced Echelon Form ........................ 46 1 Gauss-Jordan Reduction ...................... 46 2 Row Equivalence .......................... 52 Topic: Computer Algebra Systems ................... 62 Topic: Input-Output Analysis ...................... 64 Topic: Accuracy of Computations .................... 68 Topic: Analyzing Networks ........................ 72 Chapter Two: Vector Spaces 79 I Definition of Vector Space ...................... 80 1 Definition and Examples ...................... 80 2 Subspaces and Spanning Sets ................... 91 II Linear Independence ......................... 102 1 Definition and Examples ...................... 102 III Basis and Dimension ......................... 113 1 Basis ................................. 113 2 Dimension .............................. 119 3 Vector Spaces and Linear Systems ................ 124 4 Combining Subspaces¤ ....................... 131 Topic: Fields ................................ 141 Topic: Crystals .............................. 143 Topic: Voting Paradoxes ......................... 147 Topic: Dimensional Analysis ....................... 152 vii Chapter Three: Maps Between Spaces 159 I Isomorphisms ............................. 159 1 Definition and Examples ...................... 159 2 Dimension Characterizes Isomorphism .............. 168 II Homomorphisms ........................... 176 1 Definition .............................. 176 2 Rangespace and Nullspace ..................... 183 III Computing Linear Maps ....................... 195 1 Representing Linear Maps with Matrices ............. 195 2 Any Matrix Represents a Linear Map¤ .............. 205 IV Matrix Operations .......................... 212 1 Sums and Scalar Products ..................... 212 2 Matrix Multiplication ....................... 214 3 Mechanics of Matrix Multiplication ................ 222 4 Inverses ............................... 231 V Change of Basis ............................ 238 1 Changing Representations of Vectors ............... 238 2 Changing Map Representations .................. 242 VI Projection ............................... 250 1 Orthogonal Projection Into a Line¤ ................ 250 2 Gram-Schmidt Orthogonalization¤ ................ 254 3 Projection Into a Subspace¤ .................... 260 Topic: Line of Best Fit .......................... 269 Topic: Geometry of Linear Maps .................... 274 Topic: Markov Chains .......................... 281 Topic: Orthonormal Matrices ...................... 287 Chapter Four: Determinants 293 I Definition ............................... 294 1 Exploration¤ ............................ 294 2 Properties of Determinants .................... 299 3 The Permutation Expansion .................... 303 4 Determinants Exist¤ ........................ 312 II Geometry of Determinants ...................... 319 1 Determinants as Size Functions .................. 319 III Other Formulas ............................ 326 1 Laplace’s Expansion¤ ........................ 326 Topic: Cramer’s Rule ........................... 331 Topic: Speed of Calculating Determinants ............... 334 Topic: Projective Geometry ....................... 337 Chapter Five: Similarity 349 I Complex Vector Spaces ........................ 349 1 Factoring and Complex Numbers; A Review¤ .......... 350 2 Complex Representations ..................... 351 II Similarity ............................... 353 viii 1 Definition and Examples ...................... 353 2 Diagonalizability .......................... 355 3 Eigenvalues and Eigenvectors ................... 359 III Nilpotence ............................... 367 1 Self-Composition¤ ......................... 367 2 Strings¤ ............................... 370 IV Jordan Form .............................. 381 1 Polynomials of Maps and Matrices¤ ................ 381 2 Jordan Canonical Form¤ ...................... 388 Topic: Method of Powers ......................... 401 Topic: Stable Populations ........................ 405 Topic: Linear Recurrences ........................ 407 Appendix A-1 Propositions ............................... A-1 Quantifiers ............................... A-3 Techniques of Proof .......................... A-5 Sets, Functions, and Relations ..................... A-7 ¤Note: starred subsections are optional. ix Chapter Five Similarity While studying matrix equivalence, we have shown that for any homomorphism there are bases B and D such that the representation matrix has a block partial- identity form. µ ¶ Identity Zero Rep (h) = B;D Zero Zero ~ ~ ~ This representation describes the map as sending c1¯1 + ¢ ¢ ¢ + cn¯n to c1±1 + ~ ¢ ¢ ¢ + ck±k + ~0 + ¢ ¢ ¢ + ~0, where n is the dimension of the domain and k is the dimension of the range. So, under this representation the action of the map is easy to understand because most of the matrix entries are zero. This chapter considers the special case where the domain and the codomain are equal, that is, where the homomorphism is a transformation. In this case we naturally ask to find a single basis B so that RepB;B(t) is as simple as possible (we will take ‘simple’ to mean that it has many zeroes). A matrix having the above block partial-identity form is not always possible here. But we will develop a form that comes close, a representation that is nearly diagonal. I Complex Vector Spaces This chapter requires that we factor polynomials. Of course, many polynomials do not factor over the real numbers; for instance, x2 + 1 does not factor into the product of two linear polynomials with real coefficients. For that reason, we shall from now on take our scalars from the complex numbers. That is, we are shifting from studying vector spaces over the real numbers to vector spaces over the complex numbers — in this chapter vector and matrix entries are complex. Any real number is a complex number and a glance through this chapter shows that most of the examples use only real numbers. Nonetheless, the critical theorems require that the scalars be complex numbers, so the first section below is a quick review of complex numbers. 349 350 Chapter Five. Similarity In this book we are moving to the more general context of taking scalars to be complex only for the pragmatic reason that we must do so in order to develop the representation. We will not go into using other sets of scalars in more detail because it could distract from our goal. However, the idea of taking scalars from a structure other than the real numbers is an interesting one. Delightful presentations taking this approach are in [Halmos] and [Hoffman & Kunze]. I.1 Factoring and Complex Numbers; A Review This subsection is a review only and we take the main results as known. For proofs, see [Birkhoff & MacLane] or [Ebbinghaus]. Just as integers have a division operation — e.g., ‘4 goes 5 times into 21 with remainder 1’ — so do polynomials. 1.1 Theorem (Division Theorem for Polynomials) Let c(x) be a poly- nomial. If m(x) is a non-zero polynomial then there are quotient and remainder polynomials q(x) and r(x) such that c(x) = m(x) ¢ q(x) + r(x) where the degree of r(x) is strictly less than the degree of m(x). In this book constant polynomials, including the zero polynomial, are said to have degree 0. (This is not the standard definition, but it is convienent here.) The point of the integer division statement ‘4 goes 5 times into 21 with remainder 1’ is that the remainder is less than 4 — while 4 goes 5 times, it does not go 6 times. In the same way, the point of the polynomial division statement is its final clause. 1.2 Example If c(x) = 2x3 ¡ 3x2 + 4x and m(x) = x2 + 1 then q(x) = 2x ¡ 3 and r(x) = 2x + 3. Note that r(x) has
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