Review Sheet Math 450 Chapter 1
Section 1.1
Definitions: Linear vs. Non-linear Equations, Systems of Equations, Inconsistent Systems, Solution Sets, Parametric Representation of Solution Sets, Back Substitution, Elimination
Know how to: tell if a system is linear or non-linear, solve a system of equations using back substitution and elimination, write the parametric representation of a solution set to a system of equations
Suggested Problems: 1-30 p11
Section 1.2 Gaussian Elimination and Jordan Elimination
Definitions: Matrix, Subscript, Coefficient Matrix, Augmented Matrix, Elementary Row Operations, Row Echelon Form, Reduced Row Echelon Form, Gaussian Elimination with Back Substitution, Gauss - Jordan Elimination, Homogeneous System of Equations
Know how to: Find the size of a matrix, Tell if a matrix is in row echelon from, reduced row echelon form, or neither, Solve a system of equations using Gaussian Elimination with Back Substitution, Solve a system of equations using Gauss-Jordan Elimination
Suggested Problems: 1-36, pp. 26
Review Sheet Math 350 Chapter 2
Section 2.1 Operations with Matrices
Definitions: Matrix Addition, Scalar Multiplication, Matrix Multiplication, Linear Combination, Elements of Matrices
Know how to: Add and Multiply Matrices, Do Scalar Multiplication, Tell when an operation involving matrices can not be done
Suggested Problems: 1-18, 21-284 pp. 56-57
Section 2.2 Properties of Matrix Operations
Definitions: Distributive Laws, Zero Matrix, Identity Matrix, Transpose of a Matrix, Powers of a Matrix, Number of Solutions to a Linear System, Matrix Multiplication is not Commutative
Know how to: Do Algebra with Matrices.
Suggested Problems: 1-26, pp. 70-71
Section 2.3 The Inverse of a Matrix
Definitions: Matrix Inverse, Gauss-Jordan Method for finding the inverse of a matrix
Know how to: Find the inverse of a matrix
Suggested Problems: 1-20 pp. 84-85
Review Sheet Math 350 Chapter 3
Section 3.1 Determinants
Definitions: Determinant of a Matrix, Minors, Cofactors, Cofactor Expansion, Triangular Matrices, Diagonal Matrices
Know how to: Find the determinant of a matrix using cofactor expansion, also know the short cuts for finding the determinants of 2x2, 3x3, triangular, and diagonal matrices.
Suggested Problems: 1-34, pp. 130-134
Section 3.2 Evaluation of Determinants Using Elementary Row Operations
Theorems: Theorems 3.3 and 3.4, very important
Know how to: Find the determinant of a matrix using elementary row operations
Suggested Problems: 25-38, p 141
Section 3.3 Properties of Determinants
Definitions: Eigenvalues, Eigenvectors, Characteristic Equation
Theorems: Theorems 3.5,3.6,3.7,3.8,3.9 very important
Know how to: Given det(A), det(B), find det(AB), det(cA), det(AT), det(A-1), find eigenvalues and eigenvectors
Suggested Problems: 23-26, p 150
Section 3.5 Applications of Determinants
Definitions: Adjoint of a Matrix, Matrix of Cofactors, Cramer's Rule
Know how to: Find the Adjoint of a Matrix, Find the inverse of a matrix using its adjoint, solve a system of equations using Cramer's Rule
Suggested Problems: 1-2, 17-32, p. 168
Review Sheet Math 350 Chapter 4
Section 4.1 Vectors in Rn
Definitions: vector, components, vector addition, scalar multiplication, Properties of Vector Additions, linear combination
Know how to: add, subtract vectors, say if one vector is a scalar multiple of another, solve an equation for a vector
Suggested Problems: 1-32, p 188
Section 4.4 Spanning Sets and Linear Independence
Definitions: Linear Combination(p.49), Spanning Set, Span, Linear Independence, Linear Dependence, Tests for Linear Independence,
Know how to: Tell if a set is linearly independent or dependent, tell if a set spans a vector space, write a vector as a linear combination of a set of vectors
Suggested Problems: 23-34, p 219
Section 4.5 Basis and Dimension
Definitions: Basis, Dimension, Standard Basis,
Know how to: Tell if a set is a basis for a given vector space, Say why a set can not be a basis for a given vector space
Suggested Problems: 7-20, 35-42, pp. 230-231
Section 4.6 Rank of a Matrix and Systems of Linear Equations
Definitions: Rowspace, Columnspace, Nullspace, rank, nullity,
Theorems: rank(A) + nullity(A) = n, where n is the number of columns in A.
Know how to: Find a basis for the Rowspace, Columnspace, and Nullspace of a matrix, Find a basis for the span of a set of vectors, Find the rank and nullity of a matrix
Suggested Problems: 1-10, 21-32 pp. 246-247
Review Sheet Math 350 Chapter 5
Section 5.1 Length and Dot Product in Rn
Definitions: Length of a Vector, Dot Product of Two Vectors, Unit Vector, Distance Between Vectors, Orthogonal Vectors, Angle Between Vectors
Theorems: Theorems 5.1, 5.2, 5.3, 5.4 Cauchy Schwarz Inequality, 5.5 Triangle Inequality, 5.6 Pythagorean Theorem, especially 5.2, 5.4, and 5.5
Know how to: Find length of a vector, Dot product of two vectors, Angle between two vectors, Tell if two vectors are orthogonal, Find a unit vector in same direction as given vector
Suggested Problems: 1-26, 29-34, 63-72, 81-88, pp. 290-291
Section 5.2 Inner Product Spaces
Definitions: Inner Product, Norm, Distance, Angle, Orthogonal, Orthogonal Projection
Theorems: Theorems 5.7, 5.8, 5.9, especially 5.8
Know how to: Find inner product of two vectors, distance between two vectors, angle between two vectors, norm of a vector, the orthogonal projection of a vector along another vector, tell if two vectors are orthogonal
Suggested Problems: 1-16 pp. 304-305
Section 5.3 Orthonormal Bases: Gram-Schmidt Process
Definitions: Orthogonal Set, Orthonormal Set, Gram-Schmidt Process
Theorems: Theorem 5.11 Coordinates for an orthonormal basis and Theorem 5.12 Gram-Schmidt Process
Know how to: Tell if a set of vectors is orthogonal, orthonormal, or neither, Find coordinates of a vector relative to an orthonormal basis, Find an orthonormal basis using Gram-Schmidt Process
Suggested Problems: 1-14, 24-26, p 318
Last updated December 4, 2009.