Transcript chapter3
Chapter 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)
More commonly:
3 5 2 x 4
2 0 9 y 12
0 4 2 z 3
3 5 2
2 0 9
0 4 2
4
12
3
(can solve same problem
using matrix algebra tricks)
ex: Boas- see transparency.
Allowed Moves: “Row operations”
1)
2)
3)
Exchanging two rows (not columns!)
Multiply or divide a row by a nonzero constant.
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: a11 a12
a21 a22
a13
a23
a31 a32
a33
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:
a b
c d
ad bc
2) 3x3 matrix:
a11 a12
a21 a22
a31 a32
a13
11
a22
a23 ( 1) a11
a32
a33
a23
a33
1 2
( 1) a12
a21 a23
a31 a33
1 3
( 1) a13
You can do this with any row of column.
ex:
1 2 0
M 3 0 4
0 2 1
Find det(M)
*each person gets a different row or column*
a21 a22
a31 a32
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:
1 2 0
same M 3 0 4
0 2 1
Preclass Q1
B. Cramer’s Rule
Say we have a system of equations:
a1 b1 c1
a b c
2
2
2
(e.g. 2 equations and 2 unknowns.)
The solutions for x and y are:
x
det(M1 )
det(M )
y
det(M2 )
det(M )
Where
c b1
a1 b1
a1 c1
M1 1
M
M
2
3
a b
a c
2
c2 b2
2
2 2
(this generalizes any n equations with n unknowns.)
ex:
3 0 2 1
0 2 3 4
3 6 7 12
Preclass Q2
find?
III. Matrix Operations
3 2
1 2
1 3 , M 3 1 , M 1 2 3
M
Let 1
2
3
3 1 2
4 5
2 2
1) Dimension:
(# rows) x (# columns) dim (M1) = 3x2 , dim(M3) = 2x3
2) Equality:
a11
am1
a1n b11
b1n
amn bm1
bmn
iff aij bij i & j
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.
3) Transpose:
(Exchange rows and columns.)
Then
3 1 4
M
2
3
5
T
1
4) Multiplying by a scalar:
ex:
6 4
2M1 2 6
8 10
5) Adding matrices:
ex:
4 4
M1 M2 4 4
6 7
Note: can’t add M1 and M3 because they aren’t the same dimension.
6) Multiplying Matrices: (nxm matrix) x (mxn matrix) = (nxn matrix)
3 2
9 8 13
1 2 3
M1 M3 1 3
10
5
9
3
1
2
4 5
19 13 22
ex:
ijth element [M1M3]ij = Multiply row i by column j and add up terms.
7) Special Matrices:
• Unit matrix: All diagonal terms are 1, and all others are 0.
1
0
I
0
0
0
1
0
0
0
0
1
0
0
0
U
0
1
(square nxn matrix)
Note: I·M=M·I=M for any matrix M of the same dimension as I.
• Diagonal matrix:
1 2 3
0
4
5
0 0 6
Lower diagonal: 1 0 0
2 3 0
4 5 6
Upper diagonal:
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
M 1 det(1M ) CT
where C is the matrix of cofactors Cij of elements Mij.
defn: Cofactor Cij of Mij is
(-1)I+j •
determinant of matrix remaining
when row I and column j are
crossed out of M.
ex:
0 2 3
M 3 0 2
1 3 2
2
5 0
C23 ( 1)
( 1)( 2) 2
1 3
ex: Find M-1
0 2 1
M 1 0 2
1 1 2
IV. Examples
1)
3 2 1 x 3
3x + 2y + z = -3
x +
2z = 1 1 0 2 y 1
2x + y
= 4
2 1 0 z 4
A
We write Ax = b
To solve for x: Ax = b
A-1Ax = A-1b
x = A-1 b
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:
A
A
P1 Ts1
P20 Ts 20
P Ts
AP = Ts P = A-1Ts
(Matlab solves in 30 seconds.)
2) Geometry: Reflection
1 0 x x
0 1 y y
y
(-x,y)
(x,y)
x
(reflects about x-axis)
x x
? y y
(reflects about the y-axis)
3) Geometry: Rotation of coordinates
y’
y
(x’,y’)
(x,y)
x’
θ
x
x x cos y sin
y x sin y cos
cos
sin
sin x x
cos y y
(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?
4) Geometric Optics
θ2
θ1
y2
y1
Lens
(focal lenth f1)
d
Lens
(focal lenth f2)
Describe each ray by height y and angle θ.
Given (y1, θ1), what is (y2, θ2) at the output?
ex: Propagating through free space
y 2 y1 1d
1 d
M
0 1
θ2
θ1
d
y
d y
2
2
sin1
y d1
ex: Refraction at boundary
y2 y1
n1 sin1 n2 sin2
n11 n22
n1
2 1
n2
1 0
M
n
1
0
n2
y2
θ2
y1
θ1
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:
Mx x
where λ is a constant
ex: Reflection at about the y-axis
1 0
M
0
1
Eigenvectors [K1,0] , [0,K2] where K1, K2
Eigenvalues λ=-1 λ=1
(-x,y)
(x,y)
ex: Rotation
cos
M
sin
sin
cos
If θ = 180o, Eigenvectors: all [x,y], eigenvalue λ=-1
If θ = 360o, Eigenvectors: all [x,y], eigenvalue λ=1
If θ is any other value, there are no eigenvectors & eigenvalues
y
θ
(x,y)
x
More formally:
To find eigenvalues:
b
a b a
M
d
c d c
Characteristic equation:
a
b
0
c
d
(a )(d ) bc 0
Solve for eigenvalues ; then you can get the eigenvectors
So, applying this to our examples:
ex: Reflection about y-axis
0
1 0 1
M
0
1
0
1
To find the eigenvalues:
Characteristic equation:
What are the eigenvectors?
ex: Rotation
cos
M
sin
Eigenvalues:
Eigenvectors:
sin cos
sin
cos sin
cos
ex: Find eigenvalues and eigenvectors
5 0 2
M 0 3 0
2 0 5