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MAT 2401
Linear Algebra
Exam 2 Review
http://myhome.spu.edu/lauw
Info
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Tuesday 11/18
5:00-6:??pm
No Calculators
100 points
Info
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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….
Info
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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.
Info
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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
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
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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
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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
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If you were to study only ONE thing
for the exam, study this!
Eigenvalues and Eigenvectors
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How do I know I get the correct
answers?
Applications
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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
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Collinear: 3 points are collinear if and
only if
x1
x2
x3
y1 1
y2 1  0
y3 1
Applications

Cramer’s Rule: If the system
has unique solution, then
b1 a12 a13
b2 a22 a23
b a32 a33
x 3
, y
a11 a12 a13
a21 a22 a23
a31 a32 a33
a11
a21
a31
a11
a21
a31
b1 a13
b2 a23
b3 a33
, z
a12 a13
a22 a23
a32 a33
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
a11 a12 b1
a21 a22 b2
a31 a32 b3
a11 a12 a13
a21 a22 a23
a31 a32 a33
Vector Spaces
Vector Spaces
Properties of Scalar
Multiplication
Summary of Important Vector
Spaces
Possible Problems
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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
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Given SV, does S span V?
• YES – Justify
• NO – Justify
• Give an example that the system is inconsistent.
Linear Independence
Test
Possible Problems
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Given SV, 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