chapter 3: linear algebra i. solving sets of linear equations ex: solve for x, y, z. 3x + 5y + 2z =...
TRANSCRIPT
Chapter 3: Linear AlgebraChapter 3: Linear Algebra
I. Solving sets of linear equations
ex: solve for x, y, z.
3x + 5y + 2z = -4
2x + 9z = 12
4y + 2z = 3
(can solve longhand) (can solve same problem
using matrix algebra tricks)
ex: Boas- see transparency.
More commonly:
3 5 2 4
2 0 9 12
0 4 2 3
x
y
z
3 5 2 4
2 0 9 12
0 4 2 3
Allowed Moves: “Row operations”
1) Exchanging two rows (not columns!)2) Multiply or divide a row by a nonzero constant.3) Add or subtract one row from another.
ex: Pre-class assignment
ex: Circuit- see transparency and pg. 2
ex: Circuit: Find i1, i2, i3.(Halliday and Resnick, Ch. 28, 33P)
ex: Circuit (continued)
II. Determinants(only works for square matrices)
Notation:
We can extract much useful information from a matrix by boiling it down to one number called a determinant.
A. To find the determinant:
1) 2x2 matrix:
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
a b
ad bcc d
2) 3x3 matrix:
You can do this with any row of column.
ex: *each person gets a different row or column*
Find det(M)
11 12 13
22 23 21 23 21 221 1 1 2 1 321 22 23 11 12 13
32 33 31 33 31 3231 32 33
( 1) ( 1) ( 1)
a a aa a a a a a
a a a a a aa a a a a a
a a a
1 2 0
3 0 4
0 2 1
M
3) 4x4 matrix: analogy to 3x3.And so on…
Useful facts: transparency
Do examples illustrating each – base on previous example.
Can use these to simplify finding determinants.
ex:
same
Preclass Q1
1 2 0
3 0 4
0 2 1
M
B. Cramer’s RuleSay we have a system of equations:
(e.g. 2 equations and 2 unknowns.)
The solutions for x and y are:
Where
(this generalizes any n equations with n unknowns.)
1 1 1
2 2 2
a b c
a b c
1 2det( ) det( )
det( ) det( )
M Mx y
M M
1 1 1 1 1 1
1 2 32 2 2 2 2 2
c b a b a c
M M Mc b a b a c
ex: find?
Preclass Q2
3 0 2 1
0 2 3 4
3 6 7 12
III. Matrix Operations
Let
1) Dimension: (# rows) x (# columns) dim (M1) = 3x2 , dim(M3) = 2x3
2) Equality:
Note: a) Matrices must be same size (same dimension).b) This is really a set of mxn equations (aij=bij).c) Row reduction does not give equal matrices.
1 2 3
3 2 1 21 2 3
1 3 , 3 1 , 3 1 2
4 5 2 2
M M M
11 1 11 1
1 1
n n
m mn m mn
ij ij
a a b b
a a b b
iff a b i & j
3) Transpose: (Exchange rows and columns.)
Then
4) Multiplying by a scalar:
ex:
5) Adding matrices:
ex:
Note: can’t add M1 and M3 because they aren’t the same dimension.
1
3 1 4
2 3 5TM
1
6 4
2 2 6
8 10
M
1 2
4 4
4 4
6 7
M M
6) Multiplying Matrices: (nxm matrix) x (mxn matrix) = (nxn matrix)
ex:
ijth element [M1M3]ij = Multiply row i by column j and add up terms.
1 3
3 2 9 8 131 2 3
1 3 10 5 93 1 2
4 5 19 13 22
M M
7) Special Matrices:
• Unit matrix: All diagonal terms are 1, and all others are 0.
(square nxn matrix)
Note: I·M=M·I=M for any matrix M of the same dimension as I.
• Diagonal matrix:
Upper diagonal:
Lower diagonal:
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
I U
1 2 3
0 4 5
0 0 6
1 0 0
2 3 0
4 5 6
8) Inverse of a matrix: M-1 of a square matrix M is defined by M-1·M=1 and M·M-1=1.Not all matrices M have an inverse M-1.Finding M-1 is a trick! Mathematica or (tediously) by hand.
By hand:M is square, so we can find det(M).
Then
where C is the matrix of cofactors Cij of elements Mij.
defn: Cofactor Cij of Mij is (-1)I+j •
ex:
1 1det( )
TMM C
523
0 2 3
3 0 2
1 3 2
0 2( 1) ( 1)( 2) 2
1 3
M
C
determinant of matrix remaining when row I and column j are crossed out of M.
ex: Find M-1
0 2 1
1 0 2
1 1 2
M
IV. Examples
1)3x + 2y + z = -3 x + 2z = 12x + y = 4
We write Ax = bTo solve for x: Ax = b
A-1Ax = A-1b x = A-1 b
3 2 1 3
1 0 2 1
2 1 0 4
x
y
z
A x b
ex: eliz’s project
laser
Two positions: Measure Tsurf, Ths, Tamb at each. Can write down equation for each slice relating 3 temps. 20 coupled equations!Write in the form:
AP = Ts P = A-1Ts (Matlab solves in 30 seconds.)
1 1
20 20
s
s
s
P T
A
P T
A P T
2) Geometry: Reflection
(reflects about x-axis)
(reflects about the y-axis)
1 0
0 1
x x
y y
x x
y y?
(x,y)(-x,y)
x
y
3) Geometry: Rotation of coordinates
(rotates coordinates by θ)
ex: Say I reflect (3,2) about the x-axis and then the y-axis. Then what are it’s coordinates if I use a new coordinate system rotate by /6?
cos sin
sin cos
cos sin
sin cos
x x y
y x y
x x
y y
(x’,y’)y
(x,y)
x
x’
y’
θ
4) Geometric Optics
Describe each ray by height y and angle θ. Given (y1, θ1), what is (y2, θ2) at the output?
θ2
dLens (focal lenth f1)
θ1
y1y2
Lens (focal lenth f2)
1
0 1
dM
ex: Propagating through free space
θ1
d
θ2 2 1 1y y d
12 2
siny
d y
1y d
1
2
1 0
0M n
n
ex: Refraction at boundary
2 1y y
1 1 2 2
1 1 2 2
12 1
2
sin sinn n
n n
n
n
θ1
θ2
y2
y1
V. Eigenvectors & Eigenvalues
For a given operator (matrix) M, are there any vectors that are left unchanged (except for scaling the length) by M?
eg: where λ is a constant
ex: Reflection at about the y-axis
Eigenvectors [K1,0] , [0,K2] where Eigenvalues λ=-1 λ=1
Mx x
1 0
0 1M
1 2,K K
(x,y)(-x,y)
ex: Rotation
If θ = 180o, Eigenvectors: all [x,y], eigenvalue λ=-1If θ = 360o, Eigenvectors: all [x,y], eigenvalue λ=1If θ is any other value, there are no eigenvectors & eigenvalues
cos sin
sin cosM
(x,y)
x
y
θ
More formally: To find eigenvalues:
Characteristic equation:
Solve for eigenvalues ; then you can get the eigenvectors
0
( )( ) 0
a b
c d
a d bc
a b a bM
c d c d
So, applying this to our examples:
ex: Reflection about y-axis
To find the eigenvalues: Characteristic equation:
What are the eigenvectors?
1 0 1 0
0 1 0 1M
ex: Rotation
Eigenvalues:
Eigenvectors:
cos sin cos sin
sin cos sin cosM
ex: Find eigenvalues and eigenvectors
5 0 2
0 3 0
2 0 5
M