Review Session
MAT 343
Fall 2015
Matrix Properties:
Addition:
f  a  e b  f 

h  c  g d  h 
a b   e
c d    g

 
Subtraction:
f  a  e b  f 

h  c  g d  h 
a b   e
c d    g

 
Multiplication:
a b   e
c d   g


f  ae  bg

h  ce  dg
af  bh
cf  dh 
*multiplication is NOT commutative!*
Parameterization:
1.
Write the solution to the following system of equations𝑥1 + 2𝑥2 + 𝑥3 = 1
2𝑥1 − 𝑥2 + 𝑥3 = 2
4𝑥1 + 𝑥2 + 2𝑥3 = 3
3𝑥1 + 𝑥2 + 𝑥3 = 3
LU Decomposition:
2.
Review Session
MAT 343
Fall 2015
Gauss-Jordan:
3.
Vector Spaces:
4.
Linear Transformations:
5.
Review Session
MAT 343
Fall 2015
Products and Projections:
Dot Product (Scalar Product):
[𝑎, 𝑏, 𝑐] ∙ [𝑑, 𝑒, 𝑓] = [𝑎, 𝑏, 𝑐] ∙ [𝑑, 𝑒, 𝑓]𝑇 = 𝑎𝑑 + 𝑏𝑒 + 𝑐𝑓
Dot Product is 0 when vectors are orthogonal
Scalar Projection (x onto y):
𝛼=
𝑥𝑇 𝑦
||𝑦||
Vector Projection (x onto y):
𝑝 = 𝛼𝑦 =
𝑥𝑇 𝑦
𝑦
||𝑦||
1
6. Example: Find the point on the line 𝑦 = 3 𝑥 that is closest to the point (1,4)
Cross Product:
𝑖
[𝑎1 , 𝑎2 , 𝑎3 ] × [𝑏1 , 𝑏2 , 𝑏3 ] = |𝑎1
𝑏1
𝑗
𝑎2
𝑏2
𝑘
𝑎3 |
𝑏3
Least Squares:
7. Given the following data:
x
y
0
1
3
4
6
5
What is the line of best fit?
Eigenvalues:
𝜆 is an eigenvalue if the following statement is true:
(𝐴 − 𝜆𝐼)𝒙 = 0
This means:
(𝐴 − 𝜆𝐼) is a singular matrix and 𝑑𝑒𝑡(𝐴 − 𝜆𝐼) = 0
Eigenvectors are the null space of (𝐴 − 𝜆𝐼), also called the basis of the eigenspace
Review Session
MAT 343
Fall 2015
8. Find the eigenvalues and eigenvectors of the following matrices:
3
[
3
2
]
−2
Diagonalization:
An 𝑛 × 𝑛 matrix is diagonalizable when
𝑋 −1 𝐴𝑋 = 𝐷
Where X is a matrix of all the eigenvectors and D is a matrix with the corresponding
eigenvalues on the diagonals and zeroes everywhere else
Note:
𝐴𝑘 = 𝑋𝐷𝑘 𝑋 −1
9. Diagonalize the following matrix
3 −1 −2
𝐴 = [2 0 −2]
2 −1 −1