Topics for Exam 2 - Math 21b - Fall 2000

Section of text	Additional details
3: Subspaces of Rn and their dimension
3.1: Image and kernel of a linear transformation.	Define image of a linear transformation, span of a set of vectors, subspace, kernel of a linear transformation. Image of a linear transformation is a subspace of the codomain and the kernel is a subspace of the domain. Rank is the dimension of the image.
3.2: Subspaces of Rn; Bases and linear independence
3.3: The dimension of a subspace of Rn	Unique representation relative to a basis. Rank-nullity Theorem.
4: Orthogonality and least squares
4.1: Orthonormal bases and orthogonal projections	Orthogonality, length, unit vectors, orthonormal vectors, orthonormal basis, orthogonal complement, orthogonal projection, Pythagorean Theorem in Rn, Cauchy-Schwartz inequality, angle between vectors
4.2: Gram-Schmidt process (and QR factorization)	Constructing a orthonormal basis for a subspace out of another basis.
4.3: Orthogonal transformations and orthogonal matrices	Transpose of a matrix, matrix of an orthogonal projection (BBT where columns of B are an orthonormal basis for the image V of the projection)
4.4: Least squares and data fitting	Least squares approximation, least squares solutions for an inconsistent linear system Ax = b given by solutions of the normal equation ATAx = ATb; alternate formulation of matrix of an orthogonal projection
5: Determinants
5.1: Introduction to determinants	Patterns, permutations
5.2: Properties of the determinant	det(AT) = det(A); multilinearity (linearity in the columns, rows), elementary row operations and determinants; square matrix A is invertible if and only if det(A) ¹ 0; det(AB) = det(A) det(B) where defined; det(A-1) = (det(A))-1;  Laplace expansion for calculating determinants
5.3: Geometrical interpretations of the determinant	Determinant of an orthogonal matrix (= ± 1); rotation matrices; areas, volumes, and k-volume of a k-parallelepiped given by where the columns of A are the edge vectors of the parallelepiped; determinant as expansion factor; adjoint matrix for finding A-1, Cramer's Rule.
6: Eigenvalues and eigenvectors
6.1: Dynamical systems and eigenvectors	Phase portraits, trajectories, eigenvalues, eigenvectors, discrete dynamical systems; summing up theorem
6.2: Finding the eigenvalues of a matrix	Characteristic polynomial; trace and determinant of a square matrix; algebraic multiplicity of an eigenvalue
6.3: Finding the eigenvectors of a matrix	Eigenspace associated with an eigenvalue; geometric multiplicity of an eigenvalue, dimension of an eigenspace, geometric multiplicity £ algebraic multiplicity; basis of eigenvectors (when possible); eigenvectors associated with distinct eigenvalues are linearly independent
6.4: Complex eigenvalues and rotations	Basic algebra and geometry of complex numbers and operations; modulus and argument of a complex number; polar and exponential form of a complex number; DeMoivre’s formula; Fundamental Theorem of Algebra and the existence of complex eigenvalues; trace and determinant in terms of the eigenvalues, asymptotic stability and instability in terms of the eigenvalues
6.5: Stability	Prediction of long-term behavior in terms of eigenvalues, eigenvectors, and invariant subspaces.
7: Coordinate systems
7.1: Linear coordinate systems in Rn	Coordinates of a vector relative to a basis [x]B = S-1x; matrix of a linear transformation relative to a basis and the relationship B = S-1AS; column-by-column description of the matrix of a linear transformation relative to natural choices for bases - examples with projections, reflections, basis of eigenvectors, shears
7.2: Diagonalization and similarity	Similar matrices have the same characteristic polynomial, eigenvalues, determinant, trace, and rank; examples of linear transformations with invariant rotation-dilation subspaces, shears; phase portraits of similar matrices are "the same" except for a change in coordinates