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MTH 311: Advanced Linear Algebra Semester 1, 2020-2021

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MTH 311: Advanced Linear Algebra Semester 1, 2020-2021 MTH 311: Advanced Linear Algebra Semester 1, 2020-2021 Dr. Prahlad Vaidyanathan Contents I. Preliminaries4 1. Fields.....................................4 2. Matrices and Elementary Row Operations.................5 3. Matrix Multiplication............................. 11 4. Invertible Matrices.............................. 14 II. Vector Spaces 18 1. Definition and Examples........................... 18 2. Subspaces................................... 22 3. Bases and Dimension............................. 27 4. Coordinates.................................. 35 5. Summary of Row Equivalence........................ 40 III.Linear Transformations 44 1. Linear Transformations............................ 44 2. The Algebra of Linear Transformations................... 48 3. Isomorphism.................................. 57 4. Representation of Transformations by Matrices............... 58 5. Linear Functionals.............................. 67 6. The Double Dual............................... 72 7. The Transpose of a Linear Transformation................. 78 IV.Polynomials 83 1. Algebras.................................... 83 2. Algebra of Polynomials............................ 85 3. Lagrange Interpolation............................ 88 4. Polynomial Ideals............................... 90 5. Prime Factorization of a Polynomial..................... 99 V. Determinants 104 1. Commutative Rings.............................. 104 2. Determinant Functions............................ 105 3. Permutations and Uniqueness of Determinants............... 111 4. Additional Properties of Determinants................... 116 VI.Elementary Canonical Forms 123 1. Introduction.................................. 123 2. Characteristic Values............................. 123 2 3. Annihilating Polynomials........................... 132 4. Invariant Subspaces.............................. 141 5. Direct-Sum Decomposition.......................... 151 6. Invariant Direct Sums............................ 155 7. Simultaneous Triangulation; Simultaneous Diagonalization........ 161 8. The Primary Decomposition Theorem.................... 164 VII.The Rational and Jordan Forms 170 1. Cyclic Subspaces and Annihilators..................... 170 2. Cyclic Decompositions and the Rational Form............... 175 3. The Jordan Form............................... 189 VIII.InnerProduct Spaces 196 1. Inner Products................................ 196 2. Inner Product Spaces............................. 201 3. Linear Functionals and Adjoints....................... 212 4. Unitary Operators.............................. 218 5. Normal Operators............................... 226 IX.Instructor Notes 231 3 I. Preliminaries 1. Fields Throughout this course, we will be talking about \Vector spaces", and \Fields". The definition of a vector space depends on that of a field, so we begin with that. Example 1.1. Consider F = R, the set of all real numbers. It comes equipped with two operations: Addition and multiplication, which have the following properties: (i) Addition is commutative x + y = y + x for all x; y 2 F (ii) Addition is associative x + (y + z) = (x + y) + z for all x; y; z 2 F . (iii) There is an additive identity, 0 (zero) with the property that x + 0 = 0 + x = x for all x 2 F (iv) For each x 2 F , there is an additive inverse (−x) 2 F which satisfies x + (−x) = (−x) + x = 0 (v) Multiplication is commutative xy = yx for all x; y 2 F (vi) Multiplication is associative x(yz) = (xy)z for all x; y; z 2 F (vii) There is a multiplicative identity, 1 (one) with the property that x1 = 1x = x for all x 2 F 4 (viii) To each non-zero x 2 F , there is an multiplicative inverse x−1 2 F which satisfies xx−1 = x−1x = 1 (ix) Finally, multiplication distributes over addition x(y + z) = xy + xz for all x; y; z 2 F . Definition 1.2. A field is a set F together with two operations Addition : (x; y) 7! x + y Multiplication : (x; y) 7! xy which satisfy all the conditions 1.1-1.9 above. Elements of a field will be termed scalars. Example 1.3. (i) F = R is a field. (ii) F = C is a field with the usual operations Addition : (a + ib) + (c + id) := (a + c) + i(b + d); and Multiplication : (a + ib)(c + id) := (ac − bd) + i(ad + bc) (iii) F = Q, the set of all rational numbers, is also a field. In fact, Q is a subfield of R (in the sense that it is a subset of R which also inherits the operations of addition and multiplication from R). Also, R is a subfield of C. (iv) F = Z is not a field, because 2 2 Z does not have a multiplicative inverse. Standing Assumption: For the rest of this course, all fields will be denoted by F , and will either be R or C, unless stated otherwise. 2. Matrices and Elementary Row Operations Definition 2.1. Let F be a field and n; m 2 N be fixed integers. Given m scalars m (y1; y2; : : : ; ym) 2 F and nm elements fai;j : 1 ≤ i ≤ n; 1 ≤ j ≤ mg, we wish to find n n scalars (x1; x2; : : : ; xn) 2 F which satisfy all the following equations a1;1x1 + a1;2x2 + ::: + a1;nxn = y1 a2;1x1 + a2;2x2 + ::: + a2;nxn = y2 . am;1x1 + am;2x2 + ::: + am;nxn = ym This problem is called a system of m linear equations in n unknowns. A tuple (x1; x2; : : : ; xn) 2 n F that satisfies the above system is called a solution of the system. If y1 = y2 = ::: = ym = 0, then the system is called a homogeneous. 5 We may express a system of linear equations more simply in the form AX = Y where 0 1 0 1 0 1 a1;1 a1;2 : : : a1;n x1 y1 B a a : : : a C Bx C B y C B 2;1 2;2 2;n C B 2C B 2 C A = B . C ;X := B . C ; and Y := B . C @ . A @ . A @ . A am;1 am;2 : : : am;n xn ym The expression A above is called a matrix of coefficients of the system, or just an m × n th matrix over the field F . The term ai;j is called the (i; j) entry of the matrix A. In this notation, X is an n × 1 matrix, and Y is an m × 1 matrix. In order to solve this system, we employ the method of row reduction. You would have seen this in earlier classes on linear algebra, but we now formalize it with definitions and theorems. Definition 2.2. Let A be an m × n matrix. An elementary row operation associates to A a new m × n matrix e(A) in one of the following ways: E1: Multiplication of one row of A by a non-zero scalar: Choose 1 ≤ r ≤ m and a non-zero scalar c, then e(A)i;j = Ai;j if i 6= r and e(A)r;j = cAr;j th E2: Replacement of the r row of A by row r plus c times row s, where c 2 F is any scalar and r 6= s: e(A)i;j = Ai;j if i 6= r and e(A)r;j = Ar;j + cAs;j E3: Interchange of two rows of A: e(A)i;j = Ai;j if i2 = fr; sg and e(A)r;j = As;j and e(A)s;j = Ar;j The first step in this process is to observe that elementary row operations are reversible. Theorem 2.3. To every elementary row operation e, there is an operation e1 of the same type such that e(e1(A)) = e1(e(A)) = A for any m × n matrix A. Proof. We prove this for each type of elementary row operation from Definition 2.2. E1: Define e1 by −1 e1(B)i;j = Bi;j if i 6= r and e1(B)r;j = c Br;j 6 E2: Define e1 by e1(B)i;j = Bi;j if i 6= r and e1(B)r;j = Br;j − cBs;j E3: Define e1 by e1 = e Definition 2.4. Let A and B be two m × n matrices over a field F . We say that A is row-equivalent to B if B can be obtained from A by finitely many elementary row operations. By Theorem 2.3, this is an equivalence relation on the set F m×n. The reason for the usefulness of this relation is the following result. Theorem 2.5. If A and B are row-equivalent, then for any vector X 2 F n, AX = 0 , BX = 0 Proof. By Theorem 2.3, it suffices to show that AX = 0 ) BX = 0. Furthermore, we may assume without loss of generality that B is obtained from A by a single elementary n row operations. So fix X = (x1; x2; : : : ; xn) 2 F that satisfies AX = 0. Then, for each 1 ≤ i ≤ m, we have n X (AX)i = ai;jxj = 0 j=1 We wish to show that n X (BX)i = bi;jxj = 0 j=1 We consider the different possible operations as in Definition 2.2 E1: Here, we have (BX)i = (AX)i if i 6= r and (BX)r = c(AX)r E2: Here, we have (BX)i = (AX)i if i 6= r and (BX)r = (AX)r + c(AX)s E3: Here, we have (BX)r = (AX)i if i2 = fr; sg and (BX)r = (AX)s and (BX)s = (AX)r In all three cases, BX = 0 holds. 7 Definition 2.6. (i) An m × n matrix R is said to be row-reduced if (i) The first non-zero entry of each non-zero row of R is equal to 1. (ii) Each column of R which contains the leading non-zero entry of some row has all other entries zero. (ii) R is said to be a row-reduced echelon matrix if R is row-reduced and further satisfies the following conditions (i) Every row of R which has all its entries 0 occurs below every non-zero row. (ii) If R1;R2;:::;Rr are the non-zero rows of R, and if the leading non-zero entry of Ri occurs in column ki; 1 ≤ i ≤ r, then k1 < k2 < : : : < kr Example 2.7. (i) The identity matrix I is an n × n (square) matrix whose entries are ( 1 : i = j Ii;j = δi;j = 0 : i 6= j This is clearly a row-reduced echelon matrix.
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