algebra is the study of vectors and linear functions In broad terms, vectors are things you can add
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Basic Algebra - Stony Brook Mathematics
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Linear Algebra - UC Davis Mathematics
algebra is the study of vectors and linear functions In broad terms, vectors are things you can add
MATHEMATICAL FORMULAE Algebra 1 (a + b)2 = a2 + 2ab +
1 (a + b)2 = a2 + 2ab + b2; a2 + b2 = (a + b)2 − 2ab 2 (a − b)2 = a2 − 2ab + b2; a2 + b2 = (a
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Linear Algebra
David Cherney, Tom Denton,
Rohit Thomas and Andrew Waldron
2Edited by Katrina Glaeser and Travis Scrimshaw
First Edition. Davis California, 2013.This work is licensed under aCreative Commons Attribution-NonCommercial-
ShareAlike 3.0 Unported License.
2Contents
1 What is Linear Algebra?
91.1 Organizing Information
91.2 What are Vectors?
121.3 What are Linear Functions?
151.4 So, What is a Matrix?
201.4.1 Matrix Multiplication is Composition of Functions
251.4.2 The Matrix Detour
261.5 Review Problems
302 Systems of Linear Equations
372.1 Gaussian Elimination
372.1.1 Augmented Matrix Notation
372.1.2 Equivalence and the Act of Solving
402.1.3 Reduced Row Echelon Form
402.1.4 Solution Sets and RREF
452.2 Review Problems
482.3 Elementary Row Operations
522.3.1 EROs and Matrices
522.3.2 Recording EROs in (MjI). . . . . . . . . . . . . . . . 54
2.3.3 The Three Elementary Matrices
562.3.4LU,LDU, andPLDUFactorizations. . . . . . . . . . 58
2.4 Review Problems
613 4
2.5 Solution Sets for Systems of Linear Equations
632.5.1 The Geometry of Solution Sets: Hyperplanes
642.5.2 Particular Solution+Homogeneous Solutions
652.5.3 Solutions and Linearity
662.6 Review Problems
683 The Simplex Method
713.1 Pablo's Problem
713.2 Graphical Solutions
733.3 Dantzig's Algorithm
753.4 Pablo Meets Dantzig
783.5 Review Problems
804 Vectors in Space,n-Vectors83
4.1 Addition and Scalar Multiplication inRn. . . . . . . . . . . .84
4.2 Hyperplanes
854.3 Directions and Magnitudes
884.4 Vectors, Lists and Functions:RS. . . . . . . . . . . . . . . .94
4.5 Review Problems
975 Vector Spaces
1015.1 Examples of Vector Spaces
1025.1.1 Non-Examples
1065.2 Other Fields
1075.3 Review Problems
1096 Linear Transformations
1116.1 The Consequence of Linearity
1126.2 Linear Functions on Hyperplanes
1146.3 Linear Dierential Operators
1156.4 Bases (Take 1)
1156.5 Review Problems
1187 Matrices
1217.1 Linear Transformations and Matrices
1217.1.1 Basis Notation
1217.1.2 From Linear Operators to Matrices
1277.2 Review Problems
1294 5
7.3 Properties of Matrices
1337.3.1 Associativity and Non-Commutativity
1407.3.2 Block Matrices
1427.3.3 The Algebra of Square Matrices
1437.3.4 Trace
1457.4 Review Problems
1467.5 Inverse Matrix
1507.5.1 Three Properties of the Inverse
1507.5.2 Finding Inverses (Redux)
1517.5.3 Linear Systems and Inverses
1537.5.4 Homogeneous Systems
1547.5.5 Bit Matrices
1547.6 Review Problems
1557.7 LU Redux
1597.7.1 UsingLUDecomposition to Solve Linear Systems. . . 160
7.7.2 Finding anLUDecomposition.. . . . . . . . . . . . . 162
7.7.3 BlockLDUDecomposition. . . . . . . . . . . . . . . . 165
7.8 Review Problems
1668 Determinants
1698.1 The Determinant Formula
1698.1.1 Simple Examples
1698.1.2 Permutations
1708.2 Elementary Matrices and Determinants
1748.2.1 Row Swap
1758.2.2 Row Multiplication
1768.2.3 Row Addition
1778.2.4 Determinant of Products
1798.3 Review Problems
1828.4 Properties of the Determinant
1868.4.1 Determinant of the Inverse
1908.4.2 Adjoint of a Matrix
1908.4.3 Application: Volume of a Parallelepiped
1928.5 Review Problems
1939 Subspaces and Spanning Sets
1959.1 Subspaces
1959.2 Building Subspaces
1975 6
9.3 Review Problems
20210 Linear Independence
20310.1 Showing Linear Dependence
20410.2 Showing Linear Independence
20710.3 From Dependent Independent
20910.4 Review Problems
21011 Basis and Dimension
21311.1 Bases inRn.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
11.2 Matrix of a Linear Transformation (Redux)
21811.3 Review Problems
22112 Eigenvalues and Eigenvectors
22512.1 Invariant Directions
22712.2 The Eigenvalue{Eigenvector Equation
23312.3 Eigenspaces
23612.4 Review Problems
23813 Diagonalization
24113.1 Diagonalizability
24113.2 Change of Basis
24213.3 Changing to a Basis of Eigenvectors
24613.4 Review Problems
24814 Orthonormal Bases and Complements
25314.1 Properties of the Standard Basis
25314.2 Orthogonal and Orthonormal Bases
25514.2.1 Orthonormal Bases and Dot Products
25614.3 Relating Orthonormal Bases
25814.4 Gram-Schmidt & Orthogonal Complements
26114.4.1 The Gram-Schmidt Procedure
2 6414.5QRDecomposition. . . . . . . . . . . . . . . . . . . . . . . . 265
14.6 Orthogonal Complements
26714.7 Review Problems
27215 Diagonalizing Symmetric Matrices
27715.1 Review Problems
2816 7
16 Kernel, Range, Nullity, Rank
28516.1 Range
28616.2 Image
28716.2.1 One-to-one and Onto
28916.2.2 Kernel
29216.3 Summary
29716.4 Review Problems
29917 Least squares and Singular Values
30317.1 Projection Matrices
30617.2 Singular Value Decomposition
30817.3 Review Problems
312A List of Symbols
315B Fields
317C Online Resources
319D Sample First Midterm
321E Sample Second Midterm
331F Sample Final Exam
341G Movie Scripts
367G.1 What is Linear Algebra?
367G.2 Systems of Linear Equations
367G.3 Vectors in Spacen-Vectors. . . . . . . . . . . . . . . . . . . . 377
G.4 Vector Spaces
379G.5 Linear Transformations
383G.6 Matrices
385G.7 Determinants
395G.8 Subspaces and Spanning Sets
403G.9 Linear Independence
404G.10 Basis and Dimension
407G.11 Eigenvalues and Eigenvectors
409G.12 Diagonalization
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