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# 235_course_notes - Linear Algebra 2 Course Notes for MATH...

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Unformatted text preview: Linear Algebra 2 Course Notes for MATH 235 Edition 1.0 D. Wolczuk Copyright: D. Wolczuk, 1st Edition, 2011 Contents 1 Fundamental Subspaces 1 1.1 Bases of Fundamental Subspaces . . . . . . . . . . . . . . . . . . . . 1 1.2 Subspaces of Linear Mappings . . . . . . . . . . . . . . . . . . . . . . 7 2 Linear Mappings 10 2.1 General Linear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Rank-Nullity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Matrix of a Linear Mapping . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Inner Products 28 3.1 Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Orthogonality and Length . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 The Gram-Schmidt Procedure . . . . . . . . . . . . . . . . . . . . . . 43 3.4 General Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.5 The Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . 54 3.6 The Method of Least Squares . . . . . . . . . . . . . . . . . . . . . . 56 4 Applications of Orthogonal Matrices 62 4.1 Orthogonal Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2 Orthogonal Diagonalization . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4 Graphing Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . 78 4.5 Optimizing Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . 84 4.6 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . 87 5 Complex Vector Spaces 95 5.1 Complex Number Review . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2 Complex Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.3 Complex Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.4 Complex Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.5 Unitary Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.6 Cayley-Hamilton Theorem . . . . . . . . . . . . . . . . . . . . . . . . 122 ii Chapter 1 Fundamental Subspaces The main purpose of this chapter is to review several important concepts from Linear Algebra 1. These concepts include subspaces, bases, dimension, and linear mappings. It is important to ensure that you are very familiar with these concepts as we will be using them, along with other Linear Algebra 1 concepts, a lot in Math 235. Note that the Math 136 course notes are provided on the course website for your reference. 1.1 Bases of Fundamental Subspaces DEFINITION Fundamental Subspaces Let A be an m × n matrix. The four fundamental subspaces of A are (i) The columnspace of A . Col( A ) = { Avectorx | vectorx ∈ R n } (ii) The rowspace of A . Row( A ) = { A T vectorx | vectorx ∈ R m } (iii) The nullspace of A . Null( A ) = { vectorx ∈ R n | Avectorx = vector } (iv) The left nullspace of...
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235_course_notes - Linear Algebra 2 Course Notes for MATH...

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