MAT 2401 Linear Algebra Exam 2 Review .
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Transcript of MAT 2401 Linear Algebra Exam 2 Review .
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.
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.
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
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
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.
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.
Possible Problems
Given SV, does S span V?•YES – Justify
•NO – Justify•Give an example that the system is
inconsistent.
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.