Math 320, Spring 2015
Topics Covered in Each Lecture
|
Lecture Number |
Date |
Content |
Corresponding Reading material |
|
1 |
Feb. 10 | Vector spaces over real and complex numbers, examples Fn, F∞, and F[x]; uniqueness of the zero vector and additive inverse of a vector, 0.v=0, a.0=0, (-1).v=-v; subspace, examples | Pages 1-14 of Axler |
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2 |
Feb. 12 | Sum and direct sum of subspaces, characterization theorems for direct sum; span of a subset, Span(A) is the smallest subspace containing A. | Pages 14-22 of Axler |
|
3 |
Feb. 17 | Finite- and infinite-dimensional vector spaces, linear-independence | Pages 22-27 of Axler |
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Feb. 24 | Basis of a vector space, reduction of spanning lists to a bases, the extension of linearly independent lists to bases, and existence of bases in a finite-dimensional vector space, the dimension of a finite-dimensional vector space | Pages 27-31 of Axler |
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5 |
Feb. 26 | Properties of the dimension of a finite-dimensional vector space, its subspaces, and their sum; linear maps, their domain and range, examples | Pages 31-39 of Axler |
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6 |
Mar. 03 | Linear maps belonging to L(Fn,F); Image and inverse image of subspaces under linear maps, range and null space, characterinzation of 1-to-1 linear maps using their null space, dimension theorem and some of its applications | Pages 39-47 of Axler |
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Quiz 1 |
Mar. 04 | ||
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7 |
Mar. 05 | Matrix representations of a linear map, summation and scalar multiplication of matrices, vector space of m × n matrices, multiplication of matrices; invertible linear maps and isomorphic vector spaces | Pages 48-55 of Axler |
|
8 |
Mar. 10 | Isomorphism as an equivalence relation, classification of finite-dimensional vector spaces, Theorem: For a finite-dimensional vector space V and linear operators L:V->V with domain V, one-to-oneness is equivalent to ontoness. Ismorphism between L (V,W) and Mat(m,n,F) where n=dim(V) and m=dim(W); Motivation for eigenvalue problem, invariant subspaces, one-dim invariant subspaces, eigenvalues and eigenvectors of a linear operator. | Pages 55-58 & 75-77 of Axler |
|
9 |
Mar. 12 | Invariant subspaces associated with the eigenvalues of a linear map, geometric multiplicity of eigenvalues, linear independence of a list of eigenvectors with distinct eigenvalues, dimension as the upper bound on the number of eigenvalues, linear maps with as many distinct eigenvalues as the dimension, characterization of eigenvalues a of a linear map A in terms of non-invertibility of A-a I, the point spectrum, the existence of eigenvalues for linear maps acting in a finite-dimensional complex vector space | Pages 78-82 & 87-90 of Axler |
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Quiz 2 |
Mar. 13 | ||
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Mar. 17 | Upper-triangular matrices and matrix representation of linear maps acting in a finite-dimensional vector space | Pages 83-87 of Axler |
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Mar. 18 | Extra Lecture: Inner-product spaces, norm and distance defined by an inner product, unit and orthogonal vectors, orthonormal subsets of an inner-product space, Pythagorean theorem and Cauchy-Schwarz inequality | Pages 97-104 of Axler |
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12 |
Mar. 19 | Triangular inequality, orthonormal lists and bases, Gram-Schmidt process, existence of orthonormal bases, upper-triangular matrix representation of a linear operator in an orthonormal basis. | Pages 105-111 of Axler |
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13 |
Mar. 24 | Pre-Exam Review Session | |
|
14 |
Mar. 26 | Projection operators acting in a vector space V and the associated direct sum decompositions of V, orthogonal complement of a subspace of V and the associated direct-sum decomposition of V | Pages 111-112 of Axler |
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Exam 1 |
Mar. 28 | ||
|
15 |
Mar. 31 | Orthogonal projection operators and orthogonal direct sum decompositions, dual space of a vector space, the one-to-one correspondence between dual vector and vectors in a finite-dimensional inner-product space that is realized by the inner product (Riesz Lemma) | Pages 112-118 of Axler |
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16 |
Apr. 01 | Extra Lecture: The adjoint of a linear operator, its properties, and matrix representations. | Pages 118-121 of Axler |
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17 |
Apr. 02 | The relation between the null space and range of a linear operator with those of its adjoint; Self-adjoint operators and the reality of their eigenvalues, Lemma on expressing <Tu,w> in terms of <Tv,v> for 4 specific choices of v for complex inner-product spaces, the polarization formula, the characterization of self-adjoint operators acting in a complex inner-product space in terms of the reality of <Tv,v> for all v; A sufficient condition for a self-adjoint operator acting in a real inner-product spaces to be zero, the polarization formula for the inner products on a real vector space | Pages 120 & 127-130 of Axler |
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Spring Break |
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|
18 |
Apr. 14 | Normal operators and their characterization, the properties of the eigenvectors of a normal operator, the complex spectral theorem for normal operators in finite dimensions | Pages 130-134 of Axler |
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19 |
Apr. 16 | Statement of the spectral theorem in terms of orthogonal projection operators, the spectral representation of normal and self-adjoint operators, functions of normal and self-adjoint operators | See for example Hoffman and Kunze pages 335-337 |
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Quiz 3 |
Apr. 17 | ||
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20 |
Apr. 21 | Positive operators and their spectral representation, the existence of a unique positive square of a positive operator, isometries of an inner-product space and their basic properties | Pages 144-150 of Axler |
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21 |
Apr. 22 | Extra Lecture: Spectral theorem for isometries (unitary operators) acting in a finite-dimensional complex inner-product space; the relationship between isometries and self-adjoint operators, Matrix representation of isometries in orthonormal bases, the unitary matrices; Polar and singular-value decompositions | Pages 150-157 of Axler |
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22 |
Apr. 28 | Generalized eigenvectors, null space and range of powers of a linear operator acting in a finite-dimensional vector space, nilpotent operators, the algebraic multiplicity of eigenvalues and their equivalence to the number of time an eigenvalue appears as the diagonal entries of any upper-triangular representation of the operator | Pages 163-172 of Axler |
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23 |
Apr. 29 | Extra Lecture: A characterization theorem for diagonalizable operators; Characteristic polynomial, the Cayley-Hamilton Theorem, and its application in computing large powers of square matrices; Direct sum decomposition of a vector space into the subspaces of generalized eigenvectors of a linear operator | Pages 172-175 of Axler |
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24 |
May 05 | Review and Problem Session | |
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Quiz 4 |
May 07 | ||
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25 |
May 12 | Upper-triangular matrix representation of nilpotent operators, block-diagonal upper-triangular matrix representation of linear operators acting in a complex vector space, construction of square roots of I+N and T where N is nilpotent and T is invertible; minimal polynomial of a linear operator | Pages 175-179 of Axler |
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26 |
May 13 | Extra Lecture: Properties of the minimal polynomial of a linear operator | Pages 180-182 of Axler |
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27 |
May 14 | Jordan form of a Nilpotent operator, Jordan canonical form of a general linear operator | Pages 183-187 of Axler |
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Exam 2 |
May 18 | ||
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Make-up Exam |
May 25 |
Note: The pages from the textbook listed above may not include some of the subjects covered in the lectures.