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Linear transformation

let C in R

let u =(u1,u2) in V

let v =(v1,v2) in V

1) show cT(u) = T(cu)

2)show T(u+v) = T(u) +T(v)

if they are the same -> linear transformation

general solution

not enough lamda

find u

x = C1e^lamdatv +C2te^lamdatv + C2e^lamdatu

repeated lamda

x = C1e^lamdatv1 +C2te^lamdatv2

x = C1e^lamda1tv1 +C2e^lamda2tv2

eigen value & eigen vector

find eigen vector

plug lamda back to |A-lamda*I|=0

find eigen value

|A-lamda*I|=0

T(v) = lamda(v)

v =eigen vetor

lamda = eigen value

Ker(T)

Nullity

how many vector is in ker(T)

if it = 0, injective

Ref of A

write as system equation and parameter equation.

colA

REF of A

according to leading 1 of REF, refer to the original matrix a. those vector are the bases of col(A)

if it = codomain, surjective

rank

n-the nunber of non pivot colum if A

the nbuber of pivot column in A

dim(im(T)) = dim(colA)

Matrices

A^T : transpose of A: interchange the row and column vector in an m x n matrix A

square matrix: same member of m x n

main diagros { 1 0 0

0 1 0

0 0 1}

tr(A): trace of A: sum main diaro

Subspace

let u in W where u = (u1,u2,u3) let v in w where v = (v1,v2,v3) let Cin R

if those are equal, hence closed with respect to scalar multiplication W in a subspace of V

show CU in W

show u+v in W

Inverse of matrix A = A^-1

Invertible: detA not equal 0 not Invertible: detA = 0

[A|I] = [I|A^-1]

A^-1 = 1/det [W - y -Z x]

only work for 2 x 2 matrix

x = A^-1*b

inconsistance: no solution

consistance: have solution or infinite solution

Independent variable: free variable

dependent variable: bound variable

pivot column: column contain leading 1

RREF

1) its a row echelon matrix

2) any column that contain a leading 1 has 0 everywher else

elementary row operation

ERO

1) trade rows places (permit): change row

2) multiply every element in row by no zero scalar

3) add element of 1 row to correspinding element of another row

Laplace transforms

Unit step function

solve for y(t) by using the table

write step function for it

draw the pic from given information

use the laplace table

F(s) = interal e^-st f(t) dt from 0 to infinity

Second order DE

Using laplace transforms

L[f^n] = s^n L[f] - s^(n-1) f(0) - s^(n-2)f'(0) ... - sf^(n-2)(0)-f^(n-1)(0)

Undertermined coefficent

ay'' +by' +cy = f(t)

y = yh +yp

yp

f(t) is trig family

yp = Asint+Bcost y'p=Acost - Bsint y''p=-Asint-Bcost solve for A and B

f(t) is exp family

yp = Ae^t y'p=Ae^t y''p=Ae^t solve for A

if serpent bite: yp = Ate^t

if bite twice: yp = At^2e^t

f(t) is poly family

yp = At^2 + Bt + C y'p = 2At + B y''p =2A solve for A, B, C

yh =homogenous

using characteristic equation to sove for

electric circuits: LQ'' + RQ' + 1/CQ =V(t)

ay'' + by' +cy = 0

Simple harmonic oscillator: mx'' + bx' + kx =f(t)

use characteristic equation: ar^2 + br + c = 0

solve for discriminante : delta = b^2 - 4ac

if delta<0: y(t) = e^at(c1cos(bt)+c2sin(bt))

if delta=0: y(t) = c1e^rt + c2te^rt

if delta > 0: y(t) = c1e^r1t + c2e^r2t

First order DE of the form

System of differential equations

dx/dt = P(x,y) dy/dt = Q(x,y)

graph those. choose vnull point plug into hnull. if it <0, down arrow. choose Hnull point plug into vnull. if it < 0, left arrow.

set dy/dt = 0,solve for y : H null

set dx/dt = 0, solve for y: V null

system in 2 variables

first order

autonomous: no t involve

mixing and cooling

Newton's law of cooling

dT/dt = k(M-T)

T(t) = Toe^-kt + M(1-e^-kt)

temperature of the object

M is the medium temp

K is the constant of propotionality

dx/dt = rate in - rate out

rate out = (concentrate out) * (flow rate out)

rate in = (concentrate In) * (flow rate in)

Growth and decay

dA/dt = rA+d

A(t) = Aoe^rt + d/r(e^rt - 1)

d is the additional dollars per year from the depositor

r is interest rate compound continously

A(0) = Ao

dy/dt = Ky

y=yoe^kt

y(to) = yo

K<0, k is decay factor

K>0 , k is growth factor

Using Laplace transforms

L[f']=sL[f] - f(0) =sF(s) - f(0)

Intergrating factor

dy/dt + p(t)y = f(t)

intergrate both side respect to t

multiply both side by u(t)

u(t)=e^interal(p(t)dt

Intergrating directly

dy/dt = f(t)

If having IVP, find the value

intergrate both side

Separable equation

dy/dt = f(t)g(y)

if having IVP, find the value

if possible,solve for y in term of t

Intergrate each side

Now assume g(y) not equal 0, rewrite the equation in saparated or differential form: dy/g(y) = f(t)dt

set g(y) = 0 and solve for equalibrium solution, if any

dy/dt = f(y,t) = slope

Concative change

2nd derivative = 0

Stability

Concave up

Concave down.

Equilibrium

occur at y = k where dy/dt = 0

solution have to remain all the time

Isoclines "equal slopes"

Place where slope is the same

General solution: y = Ce^2t