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 general

-1 A x = b x = A b row reduce Linear Solutions [A| b] equations are x+ker(A) 0 Least square consistent: solution have solution

Partitioned Laplace matrices expansion Spot Row Determinants identical rows reduce or columns Upper Summing over triangular patterns

1 1 1 1 1 2 3 2 2 2 2 0 1 2 3 3 3 3 1 2 9 =? det 0 0 0 1 4 2 3 0 0 0 8 5 6 7 0 0 0 12 9 10 11 0 0 0 13 14 15 16

0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0 =? det 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0

0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0 =? det 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0

Image spanned by rank + nullety columns with = n leading 1 Image/Kernel ker(A- ) = λ kernel parametrized by free variables eigenspace

Reflection Projection Rotation Geometric Dilation Dilation maps Rotation Shear

T -1 A A T T A (A A)A if columns orthogonal T -1 T Projection x =(A A)A least square solution 2 P = P P v = (u.v) v onto one dimensional line

λ f is in X f+g is in X Linear Spaces vectors 0 is in X functions matrices

T(0) = 0 T(f) is in X Linear Maps T( λ f) = λ T(f) T(f+g) = T(f) +T(g)

x " = # x Solution: x(t) = x(0) e # t T HE MOTHER OF ODE ! S

2 x "! = -c x Solution: x(t) = x(0) cos(c t) + x ! (0) sin(c t)/c T HE FATHER OF ODE ! S

Cookbook Solution of p(D) f = g Solve the homogeneous problem p(D) f = 0 Find a special solution

How do we guess the special solution?

right hand side Try with sin(kt) A sin(kt) + B cos(kt) e k t k t Ae 1 A t A t + B t 2 2 At +Bt + C 1+sin(t) C+A sin(t)+B cos(t) If in Kernel or double At sin(kt) + B t cos(kt) roots multiply with t.

Nonlinear systems

We look at equations in the plane . x = f(x,y) . y = g(x,y)

An example . x = x(1-y) . y = y(x+y-2)

. x = x(1-y) null- clines . y = y(x+y-2) equi- librium points

. x = x(1-y) null- clines . y = y(x+y-2) equi- librium points

Jacobean matrix

Fourier analysis

π Fourier coefficients: ∫ 1 1 a = f(x) dx 0 π √ 2 π π ∫ 1 a = f(x) cos(nx) dx n π - π π ∫ 1 b = f(x) sin(nx) dx n π - π

Fourier approximation

Even Functions π cos- series ∫ 2 f(x) cos(nx) dx π 0

Odd Functions π ∫ 2 sin- series f(x) sin(nx) dx π 0

Blackboard Problem

Find the Fourier series of the following function: f (x,0) = 1 - π /3 - π /2 π π /3 π /2 - π -1

Parseval Identity 2 ||f|| = 8 8 a + ∑ a + ∑ b 2 2 2 0 n n n=1 n=1 Marc-Antoine Parseval

Partial differential equations heat equation: u = u xx t 2 heat type u = p(D ) u t equation wave equation: u = u tt xx 2 wave type equation: u = p(D ) u tt

Heat evolution b are Fourier coefficients of f(x,0) n

b are Fourier coefficients of f(x,0) ~n b are Fourier coefficients of f’(x,0) n Wave evolution

Heat Problem I u = 9u - 2u t xx u (x,0)=sin(3 x)

Wave Problem I u = 9u - 2u tt xx u (x,0)=sin(3 x) u (x,0)=0 t

Wave Problem II u = 9u - 2u tt xx u (x,0)=0 u (x,0)=sin(7 x) t