SLIDE 1
Math21b Final Review Spring 2015
Oliver Knill, May 4, 2015 selected slides
Matrices
Columns: image of basis vectors
B = S A S
- 1
n x m matrix
n rows and m columns
A B = B A
similarity in general
(AB)
- 1=B A
- 1 -1
(AB)
T=B A T T
Math21b Final Review Spring 2015 selected slides Oliver Knill, May - - PowerPoint PPT Presentation
Math21b Final Review Spring 2015 selected slides Oliver Knill, May - - PowerPoint PPT Presentation
Math21b Final Review Spring 2015 selected slides Oliver Knill, May 4, 2015 n x m matrix Columns: n rows and m image of basis columns vectors -1 -1 -1 =B A (AB) Matrices T T T =B A (AB) -1 A B = B A B = S A S similarity in
SLIDE 2
SLIDE 3
Linear equations
A x = b x = A b
- 1
consistent: have solution
Least square solution
row reduce [A| b] Solutions are x+ker(A)
SLIDE 4
Determinants
Laplace expansion
Partitioned matrices
Upper triangular
Summing over patterns
Row reduce
Spot identical rows
- r columns
SLIDE 5
2 0 1 1 3 2 1 2 9 2 5 6 1 3 7 9 10 11 4 8 12 13 14 15 16 1 2 2 1 1 2 3 3 3 1 2 3 0 0 0 0 0 0 0 0
det
=?
SLIDE 6
1 1 0 1 0 1 0 1 1 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0
det
=?
SLIDE 7
1 1 0 1 0 1 0 1 1 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0
det
=?
SLIDE 8
Image/Kernel
rank + nullety = n
Image spanned by columns with leading 1
kernel parametrized by free variables
ker(A- ) =
λ
eigenspace
SLIDE 9
Geometric maps
Projection
Reflection
Rotation
Shear Dilation
Rotation Dilation
SLIDE 10
P v = (u.v) v
Projection
A A
P = P
T
A (A A)A
T T
- 1
2
- nto one dimensional line
x =(A A)A
T
- 1
least square solution if columns orthogonal T
SLIDE 11
Linear Spaces f+g is in X λ f is in X 0 is in X
vectors functions matrices
SLIDE 12
Linear Maps
T(f) is in X
T(0) = 0
T(f+g) = T(f) +T(g)
T(λ f) = λ T(f)
SLIDE 13
THE MOTHER OF ODE!S
x " = #x
Solution: x(t) = x(0) e # t
SLIDE 14
THE FATHER OF ODE!S
x "! = -c x
Solution: x(t) = x(0) cos(c t) + x!(0) sin(c t)/c
2
SLIDE 15
Solution of p(D) f = g Cookbook
Solve the homogeneous problem p(D) f = 0
Find a special solution
SLIDE 16
How do we guess the special solution?
SLIDE 17
sin(kt)
A sin(kt) + B cos(kt)
e k t Ae
k t
1 A t A t + B t 2 At +Bt + C
2
If in Kernel or double roots multiply with t.
At sin(kt) + B t cos(kt)
Try with
1+sin(t) C+A sin(t)+B cos(t)
right hand side
SLIDE 18
Nonlinear systems
SLIDE 19
We look at equations in the plane
x = f(x,y) y = g(x,y)
. .
SLIDE 20
An example
x = x(1-y) y = y(x+y-2)
. .
SLIDE 21
null- clines equi- librium points
x = x(1-y) y = y(x+y-2)
. .
SLIDE 22
null- clines equi- librium points
x = x(1-y) y = y(x+y-2)
. .
SLIDE 23
Jacobean matrix
SLIDE 24
Fourier analysis
SLIDE 25
SLIDE 26
Fourier coefficients:
∫
1 π-π π f(x) cos(nx) dx a =
n
∫
1 π -π f(x) sin(nx) dx b =
n
∫
1 π π f(x) dx a =
1 √2
π π
SLIDE 27
Fourier approximation
SLIDE 28
Even Functions
cos- series ∫
2 π 0 π f(x) cos(nx) dx
SLIDE 29
Odd Functions
sin- series
∫
2 π 0 π f(x) sin(nx) dx
SLIDE 30
Blackboard Problem
SLIDE 31
Find the Fourier series of the following function: f (x,0) = π
- π
π/2
- π/2
1
- 1
π/3
- π/3
SLIDE 32
Parseval Identity
a + ∑ a + ∑ b
n=1
8
2 2 n n 2 n=1
8
||f|| =
2
Marc-Antoine Parseval
SLIDE 33
Partial differential equations
heat equation:
wave equation:
u = p(D ) u u = p(D ) u
t tt 2 2
u = u
t xx tt xx
heat type equation
wave type equation:
u = u
SLIDE 34
Heat evolution
b are Fourier coefficients of f(x,0) n
SLIDE 35
Wave evolution
b are Fourier coefficients of f(x,0) b are Fourier coefficients of f’(x,0) ~n n
SLIDE 36
u (x,0)=sin(3 x)
u = 9u - 2u
xx t
Heat Problem I
SLIDE 37
u (x,0)=sin(3 x)
u = 9u - 2u
xx tt
u (x,0)=0
t
Wave Problem I
SLIDE 38
u (x,0)=0
u = 9u - 2u
xx tt
u (x,0)=sin(7 x)
t