Section

Content

Suggested Problems

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

Section 5.5

 Orthogonal set
 Orthonormal set
 Theorem 5.5.1
 Theorem 5.5.2 Coordinates computations for orthonormal bases
 Orthogonal matrix
 Properties of orthogonal matrices
 Theorem 5.5.7 Projection onto a subspace with an orthonormal basis

1, 2, 3, 21

Section 5.6

 Theorem 5.6.1 GramSchmidt Orthogonalization Process
 Theorem 5.6.2 GramSchmidt QR Factorization
 Theorem 5.6.3 Least squares solution for matrix A with rank n via QR factorizaton

1, 2, 3, 5, 8

Section 6.1

 Eigenvector
 Eignenvalue
 Characteristic polynomial
 Complex eigenvalues and eigenvectors of real matrices
 Compute eigenvalues and eigenvectors
 Product and sum of eigenvalues
 Eigenvalues of upper triangular matrix
 Eigenvalues of similar matrices

1, 2, 11

Section 6.3

 Diagonalizable
 Theorem 6.3.2
 Arithmetic Multiplicity vs Geometric Multiplicity

1, 2
