Ls30b Midterm 2

Be able to state and explain all three equivalent definitions of a linear function. (Note: We saw the third version of the definition in week 5.)

1) Abstract- function f: Rn→ Rk is a linear function if: 1) f commutes with scalar multiplication: f(cv)=cf(v) for all scalars c and all vectors v in Rn (domain of f) 2) f commutes with addition: f(v+w)=f(v)+f(w) for all vectors v and w in Rn Domain f is Rn, codomain is Rk, so outputs f will b k-dim vector of real #, and only input kind is n-dim vec of real # 2) The only functions of f: Rn→ Rk that are linear are functions of the form f([X Y...]= aX+bY+.. Where and a and b are constants. We do this by rewrite the vector with standard basis using scalar multiplication: f([X Y...])=X[1 0] + Y[0 1] 3) function f:Rn→ Rk is only linear if comes in the following form, where ci,j coefficients are constants: f(x)= (cijX + etc) This is because [XYZ]= Xe1 +Ye2+ Ze3, and taking that as an input to f, we can write the matrix as f(e1), f(e2) etc are the columns of the matrix

Plug vector into linear function given only matrix of linear function:

1) make sure dim make sense. Rule: kxn •n dim vector works 2) for each row of matrix, go across row and simultaneously down column vector, multiply corresponding #, add up the results. This is the entry in that row of output vector- multiplying a matrix by a vector=plug vector into linear function Use version 1 def: rewrite f(vector) as scalar multiply each # by its standard basis, plug into f and do multi and add, then use matrix columns and multiply by number to get result and add Use version 2 def: rewrite vector input as [x y z w], set equal to matrix by multiply the matrix by new xyzw vector. Then, plug values into corresponding xyzw.

Long term behavior of matrix

1) unstable & stable equilibria- like leslie matrix; bad yr go to 0 is stable. Can also b oscillatory 2) neutral equilibria: final eq value depend on initial condition, infinite eq pt\compartamental model: large # object from 1 compartment to other, like disease model 3) neutral oscillation: neutral/final oscillation depend on initial condition

Be able to compute f(~v), for any linear function f and any vector ~v, given the values of f(~e1), f(~e2), etc. (Note: There are at least two ways to do this.)

1st case: know dimensions and linear, know f(e) is what, so let [x]=x[1]=xe1, and so f(x)=f(xe1)=xf(e)=mx 2nd: [x y]=xe1+ye2, take f of function to find f(e1) and f(e2) 1) given linear function f: Rn→ Rk: 1. Info need to know is values f(e1)etc. 2. Write arbitrary input to f as [x y z]=xe1+ye2+ze3.. 3. Use abstract def of linear (1st prop 2 then prop 1) and plug values in from 1.1

Distinguish a discrete-time model from differential equation (continuous time) model.

A discrete-time model has discrete time in the form [next state]=f([current state]), where Xt+1-Xt=rate of change of X, the [next state]= Xt+1=Xt...etc. Time is discrete Rt+1=0.3Rt-0.05RtFt, Notation: [Xt+1 Yt+1 ...]=f([Xt Yt ...]) The idea of ro∆ can b replaced w/ Xt+1-Xt, which is ∆ from time t to t+1, then solve for Xt+1 In a differential/continuous time equation, [∆ vector]=f([current state]), which is equal to X'=[inc]-[dec]=f(x). Time is continuous R'=0.3R-0.5RF... t=current time, t+1=next time

Be able to use the abstract linear properties (commutes with addition and commutes with scalar multiplication) to determine whether or not a function is linear.

A function is linear if it satisfies the addition and scalar multiplication rule for all vectors in the state space. So if it violates the rules, definitely not linear. If it satisfies the rule for those particular vector(s), it might be linear

Be able to recognize a linear combination of any things.

A linear combination of any things is any expression that's c1v1+c2v2+...cnvn where the coefficients cn are scalars and v is anythings , even dogs Linear combination can be like f=4x3+7x2+2x+5- linear combo of x3,x2, x, 1. But NOT a linear function! Not the same thing!

Be able to easily distinguish a function that is linear from one that is not linear.

A linear function always is f([X Y Z...])=aX+bY+cZ+... where a, b, anc c are constants.

Numbers in leslie matrix mean

Above it is from (current years), horizontal is to(next years diagonal=how much this yr pop stay in same life stage next yr Above diag=later life stage contribute to earlier life stage; birth rate Below diag= 1 life stage this yr move onto later life stage next yr

Be able to convert points from (R, S)-coordinates to (X, Y )-coordinates (easy), and vice-versa (harder), if given a set of vectors defining such a coordinate system (a basis of Rn ).

According to above def of coordinates, w=[X Y Z]=Qv1+ Rv2+Sv3 where v1 and v2 v3 r the basis So to find X, Y from R, S, j plug into above formula To find R, S from X, Y, plug into formula and make system of equations, then solve it For 3 sys eqn, u can rearrange so that u plug in one var into sys and then only solve for the value of 1 var equal to a #, then plug that into other var Another way to convert is use matrices: given basis, convert add the right side of the eqn and factor out to get matrix T. it converts R,S coordinates into X,Y coord, and the columns correspond to v1 and v2 etc. Thus, [X Y]=T [R S] To convert in the other direction, do same thing except not matrix, keep sys of equations. Then, solve for R and S, and the resulting matrix of R,S sys of eqn is the matrix T-1, which converts X, Y into R, S [R S]=T-1[X Y]

Vector operations

Addition. Head to tail in drawing. Need to b same dimensions Multiplying scalars. J multiply in drawing. f(vw) doesn't work; can't multiply two vectors as input, only vector & scalar

Know that matrices represent linear functions:

Basically, matrix is the (c1,1 c1,2 etc) Any linear function f:Rn→ Rk has matrix associated to it, then matrix f is kxn matrix, which is k rows and n columns. Mi,j refers to entry in row i, column j of matrix M

j extra info on basically RS coord conversion

Basically: M=matrix in X,Y w/ eigenvalues and eigenvectors U V. Diagonalized matrix is in U,V with the matrix having eigenvalues along its diagonal

Be able to describe the behavior that comes from a pair of complex eigenvalues in a discrete-time linear model.

Complex # is form a+bi, a and b r any real # and i is -1. j bi is an imaginary #. A is called real part, b is called impaginary part (imaginary part DOES NOT include i) If b2-4ac in quadratic is (-), both roots will b complex # complex conjugates; always same real part, but imaginary parts r negatives of each other Neutral eq pt linear model Stable spiral eq pt: vector getting shorter exp Rotate around and distance to origin=exp decay For discrete time linear model Eigenvalue thatre complex # r associated w/ rotating (spiraling in/out) behavior Abs value complex #: draw axes w/ real vs imaginary, find distance : a2+b2=c2, where c is the abs value of c=a+bi, and j plug in a and b. Same value for complex conjugate Pair of complex eigenvalues w/ abs value >1 means solution will spiral outward 9rotating combined w/ exp growth) Pair complex eigenvalues w/ abs value <1 means solution spiral in (rotate combo w/ exp growth) Pair complex eigenvalues abs value equal to 1 means solution will oscillate (neutral eq pt/center, neither exp growth/decay. rare)

Be able to compute the matrix of f ◦ g if given the matrix of f and the matrix of g. (Multiplying a matrix times another matrix.)

Composition of function F's output plugged into g's input: g(f(x)=gof. Visualize: → f(x)--> g(f(x)) Fog only work if domain of f is same as codomain g. If f:X→ Y and g: Y→ , only gof makes sense; domain of f contain range of g Math notation is reverse of visualize it; if start w x and plug into f then g, it's gof

Be able to identify which eigenvalue of a matrix is the dominant one, and what this means about the behavior of a discrete-time model.

Discrete time model long term behavior is in direction of whichever axis has dominant rate/eigenvalue Dominant eigenvalue is one whose abs value is the greatest

Be able to write a discrete-time model, given a set of assumptions or a compartment diagram.

Each compartment is a variable. Given a percentage, multiply it by the stage it's coming from, and then add to the equation of the state it's going to. The head of the arrow is going to/adding to that state, the tail is going away/subtracting from that state. In a discrete-time model, the next state is determined by the current state. So, start with the X' etc eqn= blank. Then, replace it w/ Xt+1-Xt=[inc]-[dec], and then add the Xt to the other side. Then, plug in the Xt value to get the Xt+1 value, which will then be the next input to the function. ex) per capita birth rate 30% a year, death rate prop to R2 by 0.05: Rt+1-Rt=0.3Rt-0.05Rt2→ Rt+1=1.3Rt-0.05Rt2

Be able to state the definition of eigenvalues and eigenvectors.

Eigenvector: given linear function f: Rn→ Rn, an n-dim vector v is this for f w/. So if v is eigenvector, it's not 0 vector and Mv=λv Eigenvalue λ if v≠0 and f(v)=λv for some scalar λ which is egen value of f corresponding to v Given n x n matrix M, an n-dim vector v is called eigenvector for M w/ eigenvalue λ if v≠0 and Mv=λv If v is eigenvector for M/f w/ eigenvalue, mean v=input and f=output, and output=scalar multiple of input, and input and output have to b same dimension Rn→ Rn so has to be q square matrix! Doesn't make sense for nonsquare matrices bc then input can't b scalar multiple of output Matrix n xn and eigenvector has to b n var then

linear stability analysis

Find by draw graph, draw slope at that eq pt Compute stability of eq pt : 1) compute derivative of R hand side of diff eqn w/ respect to X: df/dx Plug in eq pt X* into df/dx|x* derivative: if <0 then stable eq pt, if >0 then unstable, if =0 then inconclusive 4 types stability: stable, unstable, 2 dem semi stable

Be able to explain the biological meaning of all of the numbers that appear in the matrix of any discrete-time linear model, and which of those numbers must be 0.

For any math model, stand basis en is state=consisting of 1 unit of 1 sta var Linear stage based model, vector f(en) is the column, represent on avg 1 individual from 1 life stage contribution to all life stage the following year So number in matrix specify how much 1 ind from some life stage (column) in 1 yr will contribute to some life stage in the next yr

Be able to completely describe the behavior, in detail, of any diagonalizable discrete-time linear model just by looking at its eigenvalues and eigenvectors, by using the (R, S)-coordinate system defined by its eigenvectors.

For discrete time linear model, if u use eigenvectors of the model to define a new R, S coord sys, then once u look @ sta spa using these new coord, the sys behave exactly like a diagonal/decoupled linear model w/ [Rt+1 St+1]= (λ1 0) (0 λ2)[Rt St] Entries on diagonal r eigenvalues of the model This process is called diagonalizing the matrix/linear model So apply what know about dom eigenvalue to any discrete time linear model: long term behavior is exp growth/decay depending on dom eigenvalue (largest abs value whether >1 or <1), will b parallel to dom eigenvector line (one corresponding to dom eigenvalue)

to find long term behavior

For long term behavior, find dominant eigenvalue, which is largest abs value for discrete time model

For diagonalizing w/ complex eigenvalues

Form matrix T with vector as usual, and diagonlaize M as usual w/ Mv=λv or j by looking at λ. Separate eigenvector into real part and imaginary part , real part and then separate i out like i[0 1]. Use real and imaginary part (say real is x and imaginary number is y) to form matrix T which is (x y), x y make up column, and then get D from T-1MT In general, for λ=a+bi, matria (a b)(-b a)= a2+b2a2+b2(a b-b a), then multiply by bottom quadratic to get a2+b2(a/λ b/λ-b/λ a/λ)= λcos () + sin-sin()+cos()where is the pt a+bi in the complex plane Basically, given the Rt+a=MRt, factor out the quadratic to get the last formula where u j plug in the quadratic for the wavelength, and u see that it will rotate In R, S coord, might b a simple spiral, in X,Y, the spiral appears flattened or slanted at an angle since along R S axes Even for matrice bigger than 2x2, still separate into real and imaginary part and then the T matrix is each part lined up as a column. then matrix D is again diagonal but with squares for the complex, where the squares take the form (a b-b a) This square correspond to plane in sta spa of two new axes w/ rotation spiral in/out correspond to behavior in λn eigenvalue (λ inc from top L to bottom R) Basic formula: when deal w/ complex eigenvalue λ=a+bi 1) comput corresponding eigenvector v, which will have complex # in it 2) rewrite v in form x+iy, where x and y contain only real # 3) Use x and y in ur basis (to defines R, S coord) in place of v and its conjugate 4) when rewrite model in terms of this coord sys by diagonalize the matrix, the results almost diagonal but w/ 2x2 block (a b-b a) along the diagonal where complex eigenvalue would have been This 2x2 matrix of form (a b-b a) scales vector by factor of λ=a2+b2 rotates vector clockwise by some angle, specifically angle b/w pt a+bi and the + real axes in complex plane

linear functions generally

Function is linear if each output is cX+cY+cZ where XYZ r input var and c is constant. Not all are straight, if straight line then need to b f:R→ R Discrete time model is linear if function that defines it is linear; each component of next state has form cXt +cYtetc Diff eqn (cont time) model is called linear if function that defines it is linear; each ∆vec component is cX+cY+cZ et 1st simplest case f:R1-->R1 Key step: [x]=x[1], write as scalar multiplication, let [m]=f([1]) So f[x]=f[x•[1]]=xf[1]=x[m]=[mX] So f(x)=mX is linear, m is constant f=mx+b is NOT linear, but affine linear constant, or linear + constant 2nd simplest f:R2→ R1. only functions linear is f([X Y])=aX+bY Key step: rewrite [X Y] as X[ 1 0] + Y[0 1], so f=f(x[1 0])+f(Y[0 1])=xf[1 0] +yf[0 1]= Xa + Yb

Be able to compute f(~v) if given the matrix of f and a vector ~v. (Multiplying a matrix times a vector.)

Function linear only if can be written as f(v)=Mv for matrix M

Know how to define a new coordinate system ((R, S)-coordinates) if given a set of vectors (a basis of Rn)

Given 2x2 diagonal matrix, check if the diagonals r the eigenvalues, so if eigenvectors are [1 0] and [0 1] en is an eigenvector w/ eigenvalue λn bc mv=λv For a diagonal matrix its eigenvalues r exactly the entries along the diagonal. The eigenvector corresponding to each λn is en Eigenvalues lines of a diagonal matrix r exactly the X and Y axes. Eigenvector lines of the black bear model, which is non diagonal, r the eigenvectors! New axes; the eigenvectors r the new axes In XY plane, can use scalar multiple vectors of e and multiply it to find pt/state vector To draw in new plane, j draw in parallel line each addition of the vector; line spaces of both axes may not b equal A basis for Rn is list of n vectors (v1, v2,...vn) for which: Any vector w in Rn, possible to write w as linear combo of v1,v2,...vn. Aka, for any w, there r some scalars c for which w=c1v1+c2v2...cnvn. So can write vector as linear combo of those vectors, and that list of vectors is Rn basis In expression above, scalars c r called the coordinates of w w/ respect to this basis Geometrically, means possible to reach any pt in Rn by going some distance parallel to v1, followed by some distance parallel to v2 etc

Be able to perform basic calculations using the "abstract definition" of a linear function (the one about commuting with addition and commuting with scalar multiplication).

Given the information, see how the different inputs relate to each other- if input (in) 1=in2+in3, then if the function is linear, you can commute with addition: f(in1)=f(in2+in3)=f(in2)+f(in3), and you can commute with scalar multiplication: in1•3=in2, then f(in2)=f(in1•3)=3f(in1)

Be able to determine when it is valid to compose two functions, and therefore when it is valid to multiply two matrices.

If both f and g r linear, composition is also linear due to 2 properties of linear Since linear, have corresponding matrix To get column i of matrix BA, multiply matrix B by column i vector of matrix A; column is input v, and g(v)=Mv Multiply matrix B x A=BA, then if A=4x3 and B=2x4 it works; columns B match row A. if g=B and f=A, then composition BA is gof matrix, bc f is input to g, so output f (2nd matrix) multiply into g (1st matrix). Basically, fog and f=A, g=B, order of matrix: AB. if need to get fog(e1), multiply A x g(e1) to get that column Rule: (p x_)(_ x n) doesn't work if _≠_ Order matters! L to R; AB is NOT the same thing as BA; multi matrices is NOT commutative, but is associative. So, A(BC)=(AB)C- But order parentheses doesn't matter To find the specific # in a matrix, put A (1st matrix/input) on left, then B (2nd, output) on top and draw in matrix, then j multiply each across and add To find entry in row i column j of product matrix, go across row i of L matrix (A) and down column j of R matrix (B), multiply corresponding # and add up results Good yr followed by 2 bad followed by 2 good matrix: Mgood2Mbad2Mgood

Be able to give the size/dimensions of the matrix of f if given the domain and codomain of f, and vice versa.

If f: Rn→ Rk, then matrix dimension is k x n (k column/height, n row/width) Bc each column of matrix is f(en), which is the dimension of the output

Matrix discrete time linear model

If function define differential eqn is linear, then it's linear model. So each component of ∆ vec (x', y', etc) has form cX+cY+cZ etc Discrete time model is linear if f:{sta spa}--> {sta spa} function that defines model is linear function; each next sta Xt+1 Y t+2 has form cXt+cYt etc

Be able to describe the behavior of a discrete-time linear model if its initial state is along one of the eigenvector lines of the model.

If solution start on eigenvalue line corresponding to some eigenvalue λ for a discrete time linear model, then its behavior will exponential grow/decay along that eigenvector ling If λ>1, exp grow @ rate of λ-1 per yr If λ<1, it'll exp decay @ rate of 1-λ per yr Same rules for regular 1 var discrete time exp growth/decay So for solution like above, state @ time t is [Xt Yt]=λt[X0 Y0] Given eigenvalue and eigenvector, then axes: [J A]=1.03[J0 A0], so if initial state is a point, then next state is multiplied by the eigenvalue along that eigenvector . another way of viewing this is drawing the vector from origin and the multiply by eigenvalue.

stage based and age based model

Model of pop of single species but split pop based up on life stages, can have many var (higher dim sta spa) if more life stage Age-based model: 1 stage for each yr of life. Can have many sta var

Bifurcation in discrete time logistic model

Model: Xt+1=rXt(1-Xt). R inc→ stable oscillation→ period doubling bifurcation. attractor=distribution of points. Infinite many period doubling. Self similar

In matrix in discrete time model writing practice:

Need to write 0 in as necessary Linear if each var multiply by only constant and added. Convert discrete time to matrix: order matters! List sta var in right order so each column represents one variable's contribution In average means divide 1/that number of years, and multiply by pop Don't include time delay in discrete time model Keep life stage ones in chronological order

Be able to write down the matrix of a linear function f, if given either a formula for the function, or the values of f(~e1), f(~e2), etc.

Say 2x2 matrix [a b][c d] is matrix represent of f relative to basis (e1, e2). f([x y])= a b c d ( X Y)=aX+bYcX+dY, where the first column=f(e1) and the 2nd=f(e2) To get there, f(Xn)=f(x1e1+...)=x1f(e1)=x1(1st column)= matrix of (a11x1+a12x2 ...)=matrix • (X1 X2...) Aij is ith row and jth column For any real #: f(xyz)=f(xe1+ye2+ze3)=xf(e1)+yf(e2)+zf(e3)

Linear stability method via linear approx

Slope is rise/run so ∆f/∆x, so ∆f≈df/dx|x0•∆x when ∆x≈0 X=trajectory solution start close to eq pt X*, so m≈df/dx|x* ∆f=f(x)-f(x*)=f(x)=m∆x=x', so x'≈m(x-z) x-z=u, so x'=mu, and u'=x', so u'=mU which is valid when u≈0 Linear approx to diff eqn @x=x* is the linear diff eqn u'=mU, where u=x-x*. So if u larger, x further away from x*. U close to 0, x closer to x* Exp growth solution when m>0, u=u(0)emt so unstable eq pt at x* since x(t) start close but move away from x* u=distance b/w 1 solution and eq pt, distance exp decay to 0 when m<0 Study solution of diff vs discrete time model

Be able to completely describe the behavior of a diagonal (decoupled) discrete time linear model just by looking at its matrix.

Solution to discrete time if eqn is Xt+1=aXt is Xt=X0at If multi 2+ var in discrete time model r exp grow/decay independently of each other (ie linear w/ diagonal matrix), resulting behavior: Both rates abs value>1, then any starting sta grow awy from origin; eq pt @ origin is a source (unstable) If both abs value<1, any starting sta grow towards origins; eq pt is sink (stable) If 1 rate abs value<1 and other >1, starting sta move inward along 1 axis but outward along other, eq pt @ origin is saddle pt (unstable) If either rate is (-), behavior along corresponding axis will b exp grow/decay combo w/ fluctuating back & forth b/w + and - values along the corresponding axis If eigenvalue abs value greater than 1, matrix expand vector lying along that eigenvector. If (-), then matrix flip back and fort b/w +- value on eigenvector. If complex conjugate, matrix action is rotation.

Know what it means for a system of equations to be decoupled, and that a linear model is decoupled if and only if its matrix is diagonal.

Sys of equations (or diff equations, discrete time model) is decoupled if each eqn only contains 1 of own variables; can solve each eqn completely independent of the others, like not rlly sys of equations at all. Don't need to b linear, but only contain own variable Square nxn matrix is diagonal if all entries above and below diagonal r 0 from top L to bottom R; then linear model is decoupled! [X' Y']=(0.4 0)(0 -1.5) [X Y]. Solution: X(t)=x(0)ert, so r>0= exp grow, r<0=exp decay can b discrete time model too If r<1 then exp decay, if r>1 the nexp grow For discrete and cont/diff time linear model if matrix diagnoal/sys decoupled, each var exp grow/decay independent of other var

Matrix of stage based pop model

The e1=[10] is one juvenile, e2=[01] is one adult f(e1)=f([1 0])= f(current sta)=[next sta]=state next year if have exactly 1 juvenile this year So each column represent the state next yr resulting from the corresponding en state variable. Death rate, u have to add up all the column values and subtract from 1

Know what the standard basis vectors of Rn are (~e1, ~e2, etc)

The standard basis are the set of vectors {e1, e2, etc} of Rn where each vector ei has all 0 values except for a 1 in the ith position Writing as it: like f([c1 c2])= c1e1 + c2 e2= c1[1 0]+c2[0 1]

Be able to compute the eigenvalues of a 2 × 2 matrix, using its characteristic polynomial.

Use Mv=λv, let v=[ X Y], then plug it in to find system of equations, then solve for 1 var in eqn. Plug that var into the other eqn, and solve to separate var and λ. If one var=0, plug it in, but both values can't b 0; if both r 0, then NOT an eigenvalue/vector Solve for λ to find eigenvalue, either factor or use quadratic formula: λ=-b(-b)2-4ac2a Eqn for eigenvalues: λ2-(a+d)λ+(ad-bc), where M= a b c d . it is characteristic polynomial of M Eigenvalues of M have n roots of characteristic polynomial w/ degree n; nxn M Eigen value of 3 x 3 has 3 roots, 4x4 has 4 roots. Doesn't have to b whole # Eigenvalue can also b complex #: a real + imaginary # aka conjugate pairs If matrix has 2 real eigenvalues, can b decomposed into two 1D function using 2 new axes so M like multi along V and U axes • (*) Be able to compute an eigenvector of a matrix, given its corresponding eigenvalue. Each eigenvalue has corresponding eigenvector. Start w/ Mv=λv, replace v w/ a vector of variables (X Y Z) plug in and make it to a system of equations, and solve the sys of equations by 1 var on either side, then choose a var for 1 and wrote that eigenvector, but rlly any scalar multi for it is right For three var, same process. Solve for 1 var, plug that into other eqn, and then find an eqn w/ only 1 var on either side to solve for value. Make sure to check the work! If solving for eigenvector w/ decimal eigenvalue, might want to multiply both side by 100

Be able to determine the stability/type of the equilibrium point that's at the origin, for any discrete-time linear model, by using its eigenvalues.

Use dom eigenvalue Although all pop r dec in loggerhead sea turtle, proportion of life stage will stay same. Vector of % that give distribution of pop into various life stage is eigenvector corresponding to dom eigenvalue Linear model always eq pt @ origin, eigenvalues r entries along the diagonal/eigenvector line Can make behavior like diagonal matrix by think of new eigenvector line as axes Make diagonal matrix defining the new decoupled/diagonalized model as the new discrete time to find sys of equations. So eigenvalue determines long term along new axes Eigenvalue dont determine slope of line, slope from eigenvector The + side comes from the initial direction of the eigenvector For questions asking about long term behavior of nonlinear model, use the R,S coord to determine behavior and also to draw th emodel in

Be able to perform simple calculations using the defining equation of eigenvalues/eigenvectors (M~v = λ~v), such as computing the eigenvalue of a matrix if given its corresponding eigenvector.

Use f(v)=Mv=λv Given eigenvector and matrix, plug into eqn to find eigenvalue To check if it's an eigenvector, plug into eqn by multiply by M and see if output can b a multiple of original vector All scalar multiples of an eigenvector r also eigenvectors, by let vector Mw, and w=cv where c is scalar and v is multiple, then =cMv=cλv=λcv=λw Eigenline! Of M correspond to eigenvalue λ is the eigenvector but extended since all vectors along that line r also eigenvectors of M correspond to same eigenvalue

Be able to write any vector in Rn as a linear combination of the standard basis vectors.

Write any vector as a linear combination of standard basis vectors using the coefficients as the components/coordinates of the vector: v= [X Y Z]= Xe1+Ye2+Ze3 Visualize: ex) pt (4, -2). Is equal to 4e1 + -2e2. And since e1=[1 0] and e2= [0 1], then multiply e1 by 4 in state space and e2 by -2, then add them tip to tail and draw in new vector from origin F commutes w/ linear combinations: defining properties of linear function combined into 1: for any vector and scalar f(c1v1+c2v2)= c1f(v1)+c2f(v2) etc

How to write linear model in R, S coord

Xt+1=MXt has eigenvalues and eigenvectors (which form basis in sta spa Rn). use basis define new coord sys of R, S, and create translate matrix T,a nd inverse T-1 1) start w/ current sta in R, S coord 2) convert to standard X, Y coord using T 3) compute next state using M 4) convert result back to R, S coord using T-1 Process is called diagonalizing the matrix M Matrix D=(λ1 0) (0 λ2) = T-1MT is called diagonalization of M, and converts R S to next state. Also M=TDT-1. Eigenvalues of matrix M r the diagonal of matrix D

Use linearity and stand basis together:

given f(en) find f(XYZ), rewrite XYZ as linear combo of stand basis and there ya go j derive