Section

Content

Suggested Problems

Section 4.1

 Examples
 Properties
 L(0_{V}) = 0_{W}
 L(x) = L(x)
 Linear combinations
 Kernel of L
 Image of a Subspace under L
 Range of L
 Theorem 4.1.1

1, 4, 5, 6, 7, 9, 11, 17, 19

Section 4.2

 Theorem 4.2.1 Matrix representation of Linear Transformation from R^{n} to R^{m}
 Theorem 4.2.2 Matrix representation of Linear Transformation from V to W
 Theorem 4.3.3 Construction of Matrix A for case of V = R^{n} and W = R^{m} in Theorem 4.2.2
 Corollary Above construction via row equivalent transformation of an augmented matrix

1, 2, 3, 4, 6, 14, 18

Section 4.3

 Linear Operator
 Theorem 4.3.1
 Similarity

1, 3, 4, 5

Section 5.1

 Scalar product of vectors in R^{n}
 Length of a vector in in R^{n}
 Distance between two vectors in in R^{n}
 Angle between two vectors in R^{n}
 CauchySchwarz Inequality
 Definition of orthogonality for vectors in R^{n}
 Projections
 Vector projection of x onto y
 Scalar rojection of x onto y

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13,

Section 5.2

 Orthogonal subspaces
 Orthogonal complement of a subspace
 Theorem 5.2.1 Fundamental Subspaces Theorem
 Direct sum of subspaces
 Theorem 5.2.3

1, 2, 4, 5,

Section 5.3

 Least Square Problem for Ax = b
 Theorem 5.3.1
 Normal Equations for Least Squares Problem
 Theorem 5.3.2 Least squares soluton for matrix A with rank n

1a, 1b, 2
