Format and topics for exam 3
Math 129a

General information. Exam 2 will be a timed test of 50 minutes, covering sections 3.1-3.2, 4.1-4.3, and 5.1-5.2 of the text. No books, notes, calculators, etc., are allowed. Most of the exam will rely on understanding the problem sets (including problems to be done but not to be turned in), the quizzes, and the definitions and theorems that lie behind them. If you can do all of the homework and the quizzes, and you know and understand all of the definitions and the statements of all of the theorems we've studied, you should be in good shape.

You should not spend time memorizing proofs of theorems from the book, but you should defintely spend time memorizing the statements of the important theorems in the text, especially any named theorems.

Types of questions. All four of the previously described types of questions (computations, statements of definitions and theorems, paragraph homework-style questions, and true/false with justification) will probably appear on exam 3.

Definitions. The most important definitions and symbols we have covered are:

 3.1 determinant cofactor expansion on row i lower triangular upper triangular 4.1 subspace closed under vector addition closed under scalar multiplication zero subspace nonzero subspace null space Null A column space Col A row space Row A 4.2 basis standard basis of Rn dimension 5.1 eigenvector (linear operator) eigenvalue (linear operator) eigenvector (matrix) eigenvalue (matrix) eigenspace corresponding to eigenvalue lambda lambda-eigenvector lambda-eigenspace 5.2 characteristic equation (matrix) characteristic polynomial (matrix) characteristic equation (linear operator) characteristic polynomial (linear operator) multiplicity of an eigenvalue similar matrices

Examples. Make sure you understand the following examples.

Sect. 4.1:
Examples of non-subspaces: subset of Rn that satisfies properties 1 and 2 but not 3 of a subspace; subset that satisfies 1 and 3 but not 2. Examples of subspaces: Rn; zero subspace; solution of a homogeneous linear system; span of a finite set of vectors; column spaces, row spaces; null space of a matrix; null space of a linear transformation; range of a linear transformation. Using the definition to prove a set is a subspace (Example 4).

Theorems, results, algorithms. The most important theorems, results, and algorithms we have covered are listed below. You should understand all of these results, and you should be able to cite them as needed. You should also be prepared to recite named theorems.

Sect. 3.1:
Cofactor expansion on any row is OK (Thm. 3.1). Determinant of upper/lower triangular is diagonal multiplication (Thm. 3.2).
Sect. 3.2:
How elementary row operations affect the determinant (Thm. 3.3), A invertible if and only if detA != 0 (Thm. 3.4(a)), other properties of determinants (Thm. 3.4(b)-(d)).
Sect. 4.1:
Span of a finite set of vectors is a subspace (Thm. 4.1); null space of a matrix is a subspace (Thm. 4.2).
Sect. 4.2:
Pivot columns are basis for column space. Gaussian elimination gives basis for null space (Example 2). Reduction Principle (Thm. 4.3); Extension Principle (Thm. 4.4(a)). Every subspace has a basis (Thm. 4.4(b)). Span-Independence Theorem (new version, Thm. 1.9'); Invariance of Dimension (Thm. 4.4(c)). Two Out of Three Theorem (Thm. 4.5).
Sect. 4.3:
Subspace Size Theorem (Thm. 4.7). Dimension of column space equals rank; dimension of null space equals nullity. Nonzero rows of RREF are basis for row space (Thm. 4.6). Dimension of row space equals rank; rank of transpose equals rank.
Sect. 5.1:
lambda-eigenspace of A is null space of (A-lambdaIn).
Sect. 5.2:
Eigenvalues of A are solutions to characteristic equation/roots of characteristic polynomial. Similar matrices have same characteristic polynomial. Dimension of lambda-eigenspace is less than or equal to multiplicity of lambda (Thm. 5.1).

Types of computations. You should also know how to do the following general types of computations. (Note also that on the actual exam, there will be problems that are not of these types. Nevertheless, it will be helpful to know how to do all these types.)

Sect. 3.1:
Computing determinants by cofactor expansion. Determinant of upper/lower triangular.
Sect. 3.2:
Computing determinants by row reduction.
Sect. 4.1:
Finding spanning set for column space, row space, null space, range of linear transformation, null space of linear transformation.
Sect. 4.2:
Finding basis for column space, null space, range of linear transformation, null space of linear transformation. Using Two Out of Three Theorem to show that a set is a basis.
Sect. 4.3:
Computing dimensions of column space, null space, row space, range of linear transformation, null space of linear transformation.
Sect. 5.1:
Checking that a vector is an eigenvector. Finding basis for lambda-eigenspace.
Sect. 5.2:
Computing eigenvalues of a matrix, multiplicities of eigenvalues, bases for eigenspaces.

Not on exam. 3.1: Geometric applications of the determinant. Also, you should know the definition of determinant for practical purposes (i.e., expansion on rows), but you will not have to recite the definition. 3.2: Cramer's rule.

Good luck.

hsu@mathcs.sjsu.edu