Transcript Review
MAT 2401
Linear Algebra
Exam 2 Review
http://myhome.spu.edu/lauw
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
det( AB) det( A) det( B)
det( AB) det( BA)
1
1
det A
det A
det AT det A
det cA c n det 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
A
x1
y1
1
x
2 2
y2
x3
y3
1
x1
1
det
x2
1
2
x
3
1
y1 1
x1
1
y2 1 det x2
2
x3
y3 1
y1 1
y2 1
y3 1
Applications
Collinear: 3 points are collinear if and
only if
x1
x2
y1 1
y2 1 0
x3
y3 1
Applications
a11 x a12 y a13 z b1
a21 x a22 y a23 z b2
a x a y a z b
32
33
3
31
Cramer’s Rule: If the system
has unique solution, then
x
b1
a12
a13
a11
b1
a13
a11
a12
b1
b2
a22
a23
a21 b2
a23
a21
a22
b2
b3 a32 a33
a
b a33
a
a32 b3
, y 31 3
, z 31
a11 a12 a13
a11 a12 a13
a11 a12 a13
a21 a22 a23
a21 a22 a23
a21 a22 a23
a31
a32
a33
a31
a32
a33
a31
a32
a33
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
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.
V
W
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 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