MAT 2401Linear Algebra

Exam 2 Review

http://myhome.spu.edu/lauw

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

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Use appropriate connecting phrases/statements.

Possible problem types:•Computational

•Non computational•Recite definitions and properties

•Use properties of …

•Show that …

•etc….

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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.

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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

1

det( ) det( )det( )

det( ) det( )

1det

det

det det

det det

T

n

AB A B

AB BA

AA

A A

cA c A

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

1 11 1 1 1

2 2 2 22 2

3 3 3 3

3 3

1 11 1 1

det 1 det 12 2 2

1 1

1

1

1

x y x y

A x y x y

x y x y

x y

x y

x y

Applications

Collinear: 3 points are collinear if and only if

1 1

2 2

3 3

1

1 0

1

x y

x y

x y

Applications

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

11 12 13

21 22 2

1

23

31 32 3 33

a x a y a z

a x a y a z

a

b

bx a y a z

b

12 13 11 13 11 12

22 23 21 23 21 22

32 33 31 33 31 32

11 12 13 11 12 13 11 12 13

21 22 23 21 22 23 21 22 23

31 32 33 31 32 33 31 32 3

1 1 1

2 2 2

3 3 3

3

, ,

a a a a a a

a a a a a a

a a a a a ax y z

a a a a a a a a a

a a a a a a a a a

a a a a a a a

b b b

b b b

b

a a

b b

Vector Spaces

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

VectorAddition

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.

VW

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 SV, does S span V?•YES – Justify

•NO – Justify•Give an example that the system is

inconsistent.

Linear Independence

Test

Possible Problems

Given SV, is S linearly independent?•General approach: GJ Eliminations

Basis

Let S={v1,v2,…,vn} be a subset of a vector space V. S is called a basis for V if

1. S spans V2. 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)=nIf V={0}, then dim(V)=0