Format and topics for exam 2
Math 129a

General information. Exam 2 will be a timed test of 50 minutes, covering sections 1.6-1.7, 2.1-2.4, and 2.6-2.7 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-style questions, and true/false with justification) will probably appear on exam 2.

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

 1.6 span Span{u1,...,uk} spanning set to span (verb) 1.7 linearly dependent linearly independent homogeneous parametric representation 2.1 (matrix) product AB (i,j)-entry of AB diagonal entry diagonal diagonal matrix symmetric matrix partition of a matrix block multiplication outer product 2.2 population distribution Leslie matrix adjacency matrix 2.3 invertible matrix inverse of a matrix A-1 elementary matrix linear relationship linear correspondence property 2.6 function image domain codomain range TA transformation induced by A shear transformation linear transformation preserves vector addition preserves scalar multiplication identity transformation zero transformation standard matrix 2.7 onto one-to-one null space composition of functions

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 (e.g., the Span-Independence Theorem).

Sect. 1.6:
The Fat Matrix Theorem (Thm. 1.5), properties of span (Thm. 1.6).
Sect. 1.7:
The Thin Matrix Theorem (Thm. 1.7), linear independence of parametric representation, linear independence and linear combinations (Thm. 1.8), the Span-Independence Theorem (Thm. 1.9).
Sect. 2.1:
Basic properties of matrix multiplication (Thm. 2.2); four interpretations of matrix multiplication (p. 95).
Sect. 2.3:
Uniqueness of inverse (p. 114), basic properties of matrix inverse (Thm. 2.3, p. 116), properties of elementary matrices (p. 117), Linear Correspondence Property, proof that RREF is unique, properties of pivot columns of a matrix (Thm. 2.5).
Sect. 2.4:
The Long Theorem (Thm. 2.6), algorithm for matrix inversion (p. 128), algorithm for A-1B, invertible matrices and rank (Thm. 2.7).
Sect. 2.6:
Linearity of matrix transformations (Thm. 2.8), basic properties of linear transformations (Thm. 2.9), image of standard vectors determines linear transformation (Thm. 2.10), every linear transformation comes from a matrix (Thm. 2.11).
Sect. 2.7:
Characterization of onto linear transformations (Thm. 2.12), characterization of one-to-one linear transformations (Thm. 2.13), matrix of composite is product of matrices (Thm. 2.14), matrix of inverse is inverse of matrix (Thm. 2.14).

Types of computational problems. 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. 1.6:
Does a given set of vectors span Rn? Find smallest possible subset with the same span. Find the parametric representation of the solution set of a homogeneous system of linear equations.
Sect. 1.7:
Is a given set of vectors linearly independent? Write one vector in a set as a linear combination of others.
Sect. 2.1:
Multiply two matrices. Multiply matrices in block form.
Sect. 2.2:
Construct Leslie matrix, find stable population point. Traffic flow analysis. Interpretation of products of adjacency matrices.
Sect. 2.3:
Check if B=A-1. Invert elementary matrix.
Sect. 2.4:
Determine if matrix is invertible. Compute A-1, A-1B.
Sect. 2.6:
Domain and codomain of TA. Standard matrix of a linear transformation. Is a given transformation linear?
Sect. 2.7:
Find spanning set for range of a linear transformation. Find spanning set for null space of a linear transformation. Determine if linear transformation is onto, one-to-one.

Not on exam. 2.4: An interpretation of the inverse matrix (pp. 130-132).

Good luck.

hsu@mathcs.sjsu.edu