Math 415. Appled Linear Algebra
Final Exam Checklist

Chapter 1. Basics

This chapter contains basic material, but it is important only as it arises in different contexts in later chapters. So scan the definitions and properties and review the techniques.

Chapter 2. Vector Spaces and Bases

2.1. Real Vector Spaces

What are Rn, P(n), Mm x n and F(I) ? In particular, how are addition and scalar multiplication defined on these spaces?

2.2. Subspaces

Make sure you know your definitions and how to verify that something is a subspace.

2.3. Span and Linear Independence

Once again definitions are basic, so know what linear independence and span mean both generally and in specific applications.

2.4. Bases and Dimension

Definitions again, especially of basis and dimension and coordinates of a vector relative to a basis. It is here that Gaussian elimination will come into calculations. Be ready for example calculations in Rn and P(n)and calculation of bases for the subspace of solutions of one or more linear homogeneous equations.

2.5. The Fundamental Matrix Subspaces

The four fundamental subspaces of a matrix are mainly important for the results in Section 5.6, but you need to know their definitions and how to find bases for them since they come up in other places

In addition there are two principal superposition results in this section, Propositions 2.37 and 2.39, and you need to know what they say and how to use them and be able to replicate their proofs.

2.6. Graphs and Incident Matrices

The principal techniques here are finding a basis of independent circuits for a graph and then writing other circuits in terms of your independent ones. What is the kernel of an incidence matrix of a connected graph and why?

Chapter 3. Inner Products and Norms

3.1. Inner Products

Inner products are important for defining orthogonality of elements in Rn and P(n) and C0[a, b] and other vector spaces. You should know the definition and how to verify orthogonality, in particular, in polynomial and function spaces.

3.2. Inequalities

Know the statement of the Cauchy-Schwarz inequality be able to explain how it allows us to define angles between vectors, even if those vectors are polynomials or functions

3.4. Positive Definite Matrices

This is where our study of quadratic forms begins and it is the characterization in Theorem 3.21 that connects quadratic forms to symmetric matrices. You should know how to determine whether a symmetric matrix is positive definite, semi-definite or indefinite

3.5. Completing the Square

Completing the square is the same as the LDLT factorization of a symmetric matrix and it is here that the matrix factorizations in Chapter 1 become important. If you can complete the square, then you can more easily assess definiteness.

Chapter 4. Minimization and Least Squares Approximation

4.1. Minimization Problems

Be able to describe the closest point problem geometrically and write it down analytically as a minimization problem

4.2. Minimization of Quadratic Functions

The results in this section are fundamental to minimization problems for quadratic forms. You need to know how to take a qwadratic form you are given and write it in the form of the general quadratic function on page 186 (equation (4.10)) since Theorem 4.1 then tells you how to minimize it. Once again examples 4.2 and 4.3 are good prototype examples of how to do the minimization in practice

4.3. Least Squares and the Closest Point

What is the least squares solution of a linear system and why is it important? What are the normal equations for finding the least squares solution and how do they arise?

Chapter 5. Orthogonality


5.1. Orthogonal Bases

Here is where we begin to see the great power in working with a basis in which the vectors are orthogonal or orthonormal. Know what it means for a collection of vectors to be mutually orthogonal and how this relates to linear independence. Be able to prove linear independence for an orthogonal collection of vectors. Be able to find the coordinates of a vector relative to an orthogonal/orthonormal basis

5.2. The Gram-Schmidt Process

Not much to say here. This is an important technique.

5.3. Orthogonal Matrices

Orthogonal matrices come from putting using the vectors of an orthonormal basis as columns of a matrix. Know the definition and basic properties of orthogonal matrices (products, determinant, connection to orthonormal bases). Also be familiar with the form of rotation matrices about standard axes in 2 and 3 dimensions since these are important in Chapter 7 on linear transformations

5.5. Orthogonal Projections and Least Squares

Here is where we learn the connection between the closest point problem and orthogonal projections. Moreover, orthogonal projections are easy if the basis of your subspace is orthogonal. Review these ideas geometrically and in some examples

5.6. Orthogonal Subspaces

In many ways, the interpretations in this section are both the most theoretical and yet the most important results in the subject. Know the definition of orthogonality of two subspaces and orthogonal complements and be able to find orthogonal complements (that is, bases of them) in simple cases.

Theorem 5.55 is vital since it tells you how to deduce whether the vector b in Ax = b admits solutions, so be able to find bases of cokernels!! Moreover, Theorem 5.59 and Equation (5.82) tell us how to avoid problems with multiple solutions by looking for the unique solution of minimum norm. Then we can get the full set of solutions by adding a general vector from the kernel.

Chapter 6. Equilibrium

6.2. Electrical Networks

Chapter 7. Linearity

7.1. Linear Functions

Theorem 7.5 is a vital result since it characterizes all linear functions from Rn to Rm. as matrices so know its proof. Moreover, know how to find a matrix representation for a linear function by finding the action of the function on basis vectors.

Be familiar with the following two correspondences between linear functions and matrices: a) the matrix corresponding to a composition of functions is the product of the separate matrices, b) the matrix corresponding to the inverse of a linear function is the inverse of its own matrix representation

7.2. Linear Transformations

Know the change of basis results characterized in Eqns (7.27) and (7.28) and how to use them in practice. See Handout 3 for an example

Chapter 8. Eigenvalues

8.2. Eigenvalues and Eigenvectors

Know your definitions: eigenvalue, eigenvector, eigenspace, characteristic polynomial, multiplicity, trace, etc., and how they are interconnected. And most important, know how to solve eigenvalue problems.

8.3. Eigenvector Bases and Diagonalization

Diagonalization is one of the most important products of linear algebra. It is used everywhere in linear theory to "decouple" systems of equations. Know the leading results: a) eigenvalues corresponding to distinct eigenvalues are lin indep, b) if A is complete, then there is an invertible S and a diagonal D such that S-1AS = D. How do you find S and D? This is what we refer to as "diagonalizing A".

8.4. Eigenvalues of Symmetric Matrices

Symmetric matrices are the "best of all worlds" and we learn that here: a) eigenvalues are real, b) eigenvectors for distinct eigenvalues are orthogonal, and c) there is an orthonormal basis of Rn consisting of eigenvectors of A. Be prepared to give a proof of c), given a) and b). Consequence: how do you tell whether a symmetric matrix is positive definite (semi-definite)? Moreover, understand how this translates into the special diagonalization QTAQ = D (how do you find Q and D?). In general, recognize when your basis vectors are orthogonal and the advantages that that gives you.

Chapter 9. Eigenvalues

8.1 and 9.1. Basic Solution Techniques

Of prime importance here was the result that eigenvalues k and eigenvectors v of A give use eigensolutions u(t) = ektv to the linear system and if we can find n of these (i.e. the matrix A is complete), then the general solution is a general linear combo of these eigen solutions. Know how to implement this idea and, in particular, to solve initial value problems via eigenvalue methods

9.2. Stability of Linear Systems

Be familiar with the different definitions of equilibrium, stability and instability and how these can be assessed in linear systems using information about the eigenvalues.