MAT 2401 Linear Algebra Exam 2 Review

Info Tuesday 11/18 5:00-6:??pm No Calculators 100 points

Info Use appropriate connecting phrases/statements. Possible problem types: Computational Non computational Recite definitions and properties Use properties of … Show that … etc….

Info Use pencils and bring workable erasers. Make sure your work is neat, clear and easily readable or you will receive NO credits. Some problems may not have partial credits or “continuous spectrum” of partial credits.

Info Some problems may carry a lot of points… Be sure to pay attention to the steps of getting the answers. Most points are given to the correct process. You are supposed to know the materials from the first exam such as GJ eliminations.

Highlights

Properties of Determinants

Theorem and Consquence A square matrix A is invertible if and only if det(A)≠0. If det(A)≠0, the system AX=b has unique solution.

Eigenvalues and Eigenvectors Let A be a nxn matrix, a scalar, and x a non-zero nx1 column vector. and x are called an eigenvalue and eigenvector of A respectively if Ax= x

Steps 1. Find the characteristic equation det( I-A)=0 2. Solve for eigenvalues. 3. For each eigenvalue, find the corresponding eigenvector by using GJ eliminations.

Eigenvalues and Eigenvectors If you were to study only ONE thing for the exam, study this!

Eigenvalues and Eigenvectors How do I know I get the correct answers?

Applications Area of a Triangle

Applications Collinear: 3 points are collinear if and only if

Applications Cramer’s Rule: If the system has unique solution, then

Vector Spaces

Properties of Scalar Multiplication

Summary of Important Vector Spaces

Possible Problems Recite the 10 axioms. Given V, pinpoint why it is NOT a vector space- which one axiom it does not satisfy. Most often, give an example why this is the case.

Example 6 Z Collection of “Vectors” Scalars Vector Addition Scalar Multiplication

Subspace A nonempty subset W of a vector space V is called a subspace of V if W is a vector space under the operations of addition and scalar multiplication defined in V.

Theorem If W is a nonempty subset W of a vector space V, then W is a subspace of V if and only if 1. If u and v are in W, then u+v is in W. 2. If u is in W and c is any scalar, then cu is in W.

Linear Combination

The Span of a Subset

Spanning Set

Possible Problems Given S  V, does S span V? YES – Justify NO – Justify Give an example that the system is inconsistent.

Linear Independence

Test

Possible Problems Given S  V, is S linearly independent? General approach: GJ Eliminations

Basis Let S={v 1,v 2,…,v n } be a subset of a vector space V. S is called a basis for V if 1. S spans V 2. S is linearly independent.

Dimension of a Vector Space If a vector space V has a basis of n vectors, then n is called the dimension of V. Notation: dim(V)=n If V={0}, then dim(V)=0