par Vũ Lâm Il y a 11 années
382
Plus de détails
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
find u
x = C1e^lamdatv +C2te^lamdatv + C2e^lamdatu
x = C1e^lamdatv1 +C2te^lamdatv2
plug lamda back to |A-lamda*I|=0
|A-lamda*I|=0
v =eigen vetor
lamda = eigen value
how many vector is in ker(T)
if it = 0, injective
write as system equation and parameter equation.
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)
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
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
A^-1 = 1/det [W - y -Z x]
only work for 2 x 2 matrix
1) its a row echelon matrix
2) any column that contain a leading 1 has 0 everywher else
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
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
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
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
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
rate out = (concentrate out) * (flow rate out)
rate in = (concentrate In) * (flow rate in)
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
y=yoe^kt
y(to) = yo
K<0, k is decay factor
K>0 , k is growth factor
intergrate both side respect to t
multiply both side by u(t)
u(t)=e^interal(p(t)dt
If having IVP, find the value
intergrate both side
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
2nd derivative = 0
Stability
Concave up
Concave down.
occur at y = k where dy/dt = 0
solution have to remain all the time
Place where slope is the same