Jan 13 LectureNotes223.pdf

Jan 13 LectureNotes223.pdf - MAT223 Lecture Notes c Tyler...

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MAT223 Lecture Notes Tyler Holden, c 2017- Contents 1 Linear Systems 2 1.1 Linear Equations and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Parameterizations of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Matrix Representations of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 The Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.1 Column Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.2 The Transpose of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4.3 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.6 Matrix Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 Complex Numbers 30 2.1 Properties of the Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2 Some Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 The Polar Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 Determinants 37 3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 Properties of the Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.1 Quick Aside: Elementary Row Matrices . . . . . . . . . . . . . . . . . . . . . 43 3.3 Determinants and Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.4 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.5 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 1
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4 Vector Spaces 56 4.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2 Dot Product and Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2.1 Dot Product Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2.2 Projections onto Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3 Cross Product and Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.4 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.4.1 Span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.5 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.6 Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.7 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.7.1 Projections Onto Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.7.2 Gram-Schmidt Orthogonalization . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.8 Approximating Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2
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1 Linear Systems Linear algebra is one of the best studied and understood fields in mathematics. The amount of attention it receives is warranted by the shear extent of its applicability, in both pure mathematics and applied mathematics, physics, computer science, engineering, etc. In a very broad sense, it studies linear systems of equations, vector spaces, and linear maps between vector spaces. This course will introduce you to the fundamentals of linear algebra, with a focus on low-dimensional spaces such as R , R 2 , R 3 , though we will cover R n towards the end. I have heard linear algebra earnestly proclaimed as “the single most useful mathematics you will learn as an undergraduate.” While some people might dissent, it’s hard to overstate the utility of linear algebra. 1 Linear Systems You’ve likely seen examples of linear systems before. For example, you might have been asked to find a solution to 2 x - 3 y = - 7 - x + 2 y = 5 . This is not too difficult with only two equations and two unknowns, but what if we add more equations and more unknowns? Something along the lines of x + y + z = 4 x + 2 y + 3 z = 9 2 x + 3 y + z = 7 is much more difficult. You can imagine we could make this four equations, five equations, etc. We will develop a scheme for solving these types of systems. 1.1 Linear Equations and Systems Generally speaking, the word linear means something that respects additions and multiplication by real numbers. For example, the function f ( x ) = 2 x is linear, since f ( x + y ) = 2( x + y ) = f ( x ) + f ( y ) , and f ( cx ) = 2 cx = cf ( x ) . The word linear is used because the graph of f is precisely a straight line in the plane. Linear things play so nicely with addition and multiplication. A linear equation is any equation of the form c 1 x 1 + c 2 x 2 + · · · + c n x n = b for c 1 , c 2 , . . . , c n , b R . We refer to the c i as coefficients of the linear equation, and b as the constant term . When n = 2 this becomes the equation c 1 x 1 + c 2 x 2 = b, and the collection of x 1 and x 2 which satisfy this equation again forms a line in the plane.
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