# Lab 6 # Name: Omeed Zokaeim # I worked on this code with: # Please do all of your work for this week's lab in this worksheet. If # you wish to create other worksheets for scratch work, you can, but # this is the one that will be graded. You do not need to do anything # to turn in your lab. It will be collected by your TA at the beginning # of (or right before) next week’s lab. # Be sure to clearly label which question you are answering as you go and to # use enough comments that you and the grader can understand your code.

#43 @interact def mycoordinatefunction(r=(-10,10),s=(-10,10), target=vector([2,3])): #this creates sliders for scalar by setting up an interactive function u=r*vector ([1 ,2]) #defines the basis, our vectors u and v. v=s*vector ([1 ,.5]) wee= u+v #this calculates the sum of u + v and assigns it to our variable p=plot(u, axes_labels=["x", "y"]) + plot(v, color="red") + plot(wee, color="purple") + plot(target, color="yellow") + point(target, size=50, color="green") #plots vectors u and v, their sum, and our point, on the same set of axes through overlaying them show(p)

Interact: please open in CoCalc

#44 @interact def mycoordinatefunction(r=(-10,10),s=(-10,10), target=vector([2,3])): u=r*vector ([5,2]) #this defines a new basis by changing our vectors v=s*vector ([3,.5]) wee=[(u[0]+v[0] ,u[1]+v[1])] p=plot(u, axes_labels=["x", "y"]) + plot(v, color="red") + plot(u+v, color="purple") + list_plot(wee, size=50, color="yellow") + point(target, size=50, color="green") show(p)

Interact: please open in CoCalc

#45 @interact def mycoordinatefunction(r=(-10,10),s=(-10,10), target=vector([2,3])): u=r*vector ([1 ,1]) #define basis. v=s*vector ([2 ,2]) wee=[(u[0]+v[0] ,u[1]+v[1])] p=plot(u, axes_labels=["x", "y"]) + plot(v, color="red") + plot(u+v, color="purple") + list_plot(wee, size=50, color="yellow") + point(target, size=50, color="green") show(p) show(wee)

Interact: please open in CoCalc

#46 t=matrix(RDF, [[1, 1] ,[2, 1/2]]) #defines the 2x2 matrix t*vector([4/3, 2/3]) #this verifies the matrix's ability to convert from r,s to x,y corrdinates for point (2 ,3) t*vector([ -4 ,0]) #verifies for point ( -4 , -8) t*vector([ -2.6 ,5.6]) #verifies for point (3 , -2.4)

(2.0, 3.0)
(-4.0, -8.0)
(2.9999999999999996, -2.4000000000000004)

#I notice that the t matrix successfully converts r,s-coordinates back into the x,y coordinate system.

#47 w=matrix(RDF, [[ -1/3 ,2/3], [4/3 , -2/3]]) #thia defines a new matrix corresponding to our expanded system of equations

#48 w*vector([1 , -6])

(-4.333333333333333, 5.333333333333333)

#The interactive in #43 gives the R,S coordinates (-4.33, 5.33) which correspond to the point (1,-6) used here. This checks out.

#49 w*vector([4 ,5])

(1.9999999999999998, 2.0)

#The interactive in #43 gives the R,S coordinates (2, 2) which corresponds to the point (4,5) used here. This checks out as well, even with a very small sage rounding error.

#50 t.inverse()

[-0.3333333333333333 0.6666666666666666]
[ 1.3333333333333333 -0.6666666666666666]

#This matrix is the same as the one obtained in number 47, so the results again check out exactly.

#51 m=matrix(RDF, [[7,-4], [4,-3]]) m.eigenvectors_right()

[(5.0, [(0.8944271909999159, 0.4472135954999579)], 1), (-1.0, [(0.4472135954999579, 0.8944271909999159)], 1)]

#Eigenvector u=[1,2], a multiple of (0.447213,0.894427), has an eigenvalue of -1, and eigenvector v=[1,0.5], a multiple of (0.894427,0.447213) has an eigenvalue of 5.

#52 w*m*t

#The product of these matrices, that is their composition, gives us a diagonal matrix. The numbers in the diagonal correspond to the eigenvalues that we obtained in #51 and the anti-diagonal should have both numbers as zero (sage has a very small rounding error but the top right number is essentially zero).