Section

Content

Suggested Problems

Section 4.7

 Coordinate of a vector with respect to a basis
 Coordinate matrix
 Notation [x]_{B}
 Change of basis in R^{n}
 Transition matrix from basis B' to B
 Inverse of a transition matrix
 Theorem 4.21 Transition matrix from basis B to B'
 Coordinate representation in general vector space V

Pages 210211
5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 35, 37

Section 4.8

 Linear differential equation
 Wronskian of a set of functions
 Wronskian test for linear independence

Pages 219223
13, 14, 17, 20, 21

Section 5.1

 Length or norm of a vector in R^{n}
 Length of a scalar multiple
 Unit vector
 Distance between two vectors in R^{n}
 Dot product of two vectors in R^{n}
 Properties of the dot product
 CauchySchwarz inequality
 Angle between two nonzero vectors in R^{n} via dot product
 Definition of orthogonal
 Triangle Inequality in R^{n}
 Pythagorean Theorem in R^{n}

Pages 235236
78, 1112, 2021, 2526, 4142, 4851

Section 5.2

 Definition of inner product on a vector space
 Examples of inner products for R^{n}, M_{m,n}, C[a,b]
 Properties of inner products
 Definitions of length, distance and angle
 CauchySchwarz inequality
 Triangle Inequality
 Pythagorean Theorem
 Orthogonal Projection of a vector u on to a vector v

Pages 245247
17, 19, 21, 23, 29, 35, 39, 41, 4749, 73, 75, 7879

Section 5.3

 Orthogonal sets
 Orthonormal sets
 Orthogonal basis, Orthonormal basis
 Theorem 5.10 Orthogonal sets is linearly independent
 Corollary to Theorem 5.10, Page 251
 Theorem 5.11 Coordinates of vector with respect to an orthogonal basis
 Theorem 5.12 GramSchmidt Orthogonalization Process

Pages 257258
1, 4, 7, 10, 13, 21, 25, 27, 31, 35, 39, 41

Section 5.4

 Least Square Regression Line
 Least Squares Problem
 Orthogonal Subspaces
 Orthogonal Complement
 Direct Sum
 Properties of Orthogonal Subspaces
 Projection onto a Subspace which is the span of an orthogonal basis
 Theorem 5.15 Orthogonal Projection and Distance
 Fundamental subspaces of a matrix
 Theorem 5.16 Fundamental Subspaces of a Matrix
 Solution of the Least Squares Problem
 Normal equation of the least squares problem
 Orthogonal projections onto a subspace

Pages 269270
57, 1517, 1921, 2325, 2931, 3335

Section 5.5

 Least squares approximations in calculus
 Definition of least squares approximation
 Theorem 5.19 Least squares approximating function

Pages 282283
6365, 6971
