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Mathematics 3256
Linear Algebra
Spring 2011
Final Study Guide
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Equivalent equations and systems: box on page 5 and class notes from 1/13.
Matrix addition, scalar multiplication and transpose. See Section 1.2
Vectors and inner product / dot product. See Section 1.3
Matrix multiplication. See Section 1.3
Writing linear systems as matrix equations. Ex 16, page 29.
Properties of matrix operations. See Section 1.4
Reduced row echelon form and row echelon form of a matrix. Definition, page 62.
Recognize whether a matrix is in (reduced) row echelon form. Examples 1, 2, page 63.
Elementary row operations. Definition, page 63.
Row equivalent matrices. Definition, page 64.
Theorem 1.5, page 65. (Just the statement)
Theorem 1.6, page 68. (Just the statement)
Given a matrix, compute its reduced row echelon form.
Theorem 1.7 and Corollary 1.1, page 69. (Just the statements)
Solving linear systems: two elimination methods, page 70. Examples 9, 10, page 72.
Inconsistent and consistent systems, page 75.
Homogeneous systems, trivial and non-trivial solutions, page 75.
Theorem 1.8, page 77 (Just the statement)
The relation between the solutions to homogeneous and non-homogeneous systems,
page 77.
An inverse of a matrix.
Definition of a singular and non-singular matrix.
Uniqueness of an inverse, page 91. Theorem 1.9, page 91. (Just the statement)
Properties of inverses, page 92. Theorem 1.10, page 92. (Just the statement)
Theorem 1.11, page 94. (Just the statement)
Computing inverses using reduced row echelon form, page 94.
Using inverses of matrices to solve linear system, page 98.
Equivalent statements on page 100.
Definition of the determinant of a matrix, page 183
Determinant formulas for 3 × 3 and 2 × 2 matrices, Examples 5, 6, page 184
Properties of Determinants, Theorems 3.1 – 3.7, Corollary 3.1 (statements only), pages
181 – 189
Computing determinant by reducing a matrix to a triangular form, Example 17, page
190.
Theorem 3.8, page 191 (statement only)
Determinant of an inverse matrix, page 191
Cofactors and minors, page 196
Computing determinants using cofactor expansion, page 197.
Adjoint of a matrix, page 200
Theorem 3.11, page 201 (statement only)
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Corollary 3.3, page 202 (statement only)
Theorem 3.12, Corollary 3.4, page 203
Cramer’s Rule, page 205.
Theorem 4.2, page 230 (statement only)
Theorems 4.3, page 236 (statement only)
Cauchy-Schwarz Inequality, page 236
Determining the angle between two n-dimensional vectors, Example 10, page 237
Triangle Inequality, page 238
Linear Transformation, page 267
Properties of linear transformations, page 249.
Constructing a matrix corresponding to a linear transformation, Example 5, pages 253254
(49) Determining whether a vector is in the range of a given linear transformation.
(50) Determining whether a given map is a linear transformation.
(51) Computational problems in all assignments.
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