Math 423 - Test II Review

General Information

Test II (Thursday, April 4) will have 5 to 7 questions, some with multiple parts, and will cover sections 5.9, 6.1-6.4, 7.1-7.4 in the text. Please bring an 8½×11 bluebook. Problems will be similar to ones done for homework. You may use calculators to do arithmetic, although you will not need them. You may not use any calculator that has the capability of doing either calculus or linear algebra.

Topics Covered

Orthogonal Projections, Least Squares, and QR factorization - § 5.9.
1. Be able to find the orthogonal projection matrix Q, given an orthonormal basis for a space V₀ of R^p or C^p.
2. Be able to find the QR decomposition for a matrix and be able to use it to find solutions to simple least squares problems.
Linear Transformations and Basic Properties - § 6.1, 6.2
1. Know the definitions of these terms: linear transformation, null space (kernel), range, image, inverse, and adjoint. Be able to show that the null space and the image space are subspaces, and be able to find bases for these spaces. Be able to find inverses and adjoints in simple cases.
2. It is very important to be able to find the matrix for a linear transformation relative to bases for the domain and range. One can then solve unlikely looking problems (differential equations, for example) using this matrix. Given a linear transformation T: V-> V and a basis B for V, be able to find the matrix A for T, as well as bases for the null space and for the image space of T.
Norms and Condition numbers for Linear Transformations and Matrices - § 6.3, 6.4
1. Know how to define the norm of a linear transformation T: V-> W, where V and W are normed linear spaces. Be able to find the 1 and infinity norms for a p×q matrix A.
2. Be able to find the condition number of a matrix (1 or infinity norm). Be able to determine how stable a system of linear equations is, given the condition number of the coefficient matrix.
Eigenvalue Problems - § 7.1-7.2
1. Be able to solve eigenvalue problems. Know these terms: eigenvector, eigenvalue, eigenspace, characteristic polynomial, algebraic multiplicity, and geometric multiplicity.
2. Be able to make use of the form of the characteristic polynomial (see Theorem 7.6, pg. 285) to get information about eigenvalues. An example of this is problem 2, pg. 292. For a given eigenvalue, the geometric multiplicity is less than or equal to the algebraic multiplicity.
Linear Transformations, Eigenvalue Problems, and Matrix Representations - § 7.3
1. Recall that we defined a linear transformation T: V-> V to be diagonalizable if there is a basis relative to which the matrix of T is diagonal. Be able to show that T is diagonalizable if and only if V has a basis of eigenvectors of T.
2. For matrices, the definition above means that a p×p matrix A is diagonalizable if and only it has p linearly independent eigenvectors. Be able to diagonalize a matrix, and be able to give examples of matrices that are not diagonalizable.
Similarity - § 7.4
1. Know the definition of similar matrices: p×p matrices A and B are similar if and only if there is an invertible matrix S such that B= S^-1A S. Similar matrices effectively represent the same linear transformation, but with respect to different bases for the underlying space.
2. Be able to show that if A and B are similar, than A^k and B^k are also similar. Also, be able to calculate the matrix exponential exp(A) via diagonalizing A.
3. We are skipping the Jordan form this time around.