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# Name: Nushrat Esha # I worked on this code with: Ryan, Xavier, and Fuijia # 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.

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#46 u=vector([1,2]) v=vector([1,0.5]) T=column_matrix(RDF, [u,v]) show(T)

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$\left(\begin{array}{rr}
1.0 & 1.0 \\
2.0 & 0.5
\end{array}\right)$

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#47 R=1.35 S=0.65 T*vector([R, S]) # when verified, the value is very close to the point, with one of the numbers, 3.025, being slightly greater than 3

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(2.0, 3.0250000000000004)

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#48a # point (-5, 1.2) R=2.45 S=-7.5 T*vector([R, S])

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(-5.05, 1.1500000000000004)

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#48b # point (-4, -8) R=-4 S=0 T*vector([R, S])

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(-4.0, -8.0)

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#49 # expand the equations - (1) R = X -S (2) S = 1.5*X - 0.67*Y W=matrix(RDF, [[-0.33, 0.66], [1.33, -0.66]]) show(W)

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$\left(\begin{array}{rr}
-0.33 & 0.66 \\
1.33 & -0.66
\end{array}\right)$

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#50 W*vector([1,-6])

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(-4.29, 5.29)

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#50 check @interact def plotvectors(R=(-10,10, 0.1), S=(-10,10, 0.1), target=vector([1, -6])): u=R*vector([1,2]) v=S*vector([1,0.5]) p_u=plot(u, color="red", aspect_ratio=1.0, legend_label="u", legend_color="black") p_v=plot(v, color="green", legend_label="v", legend_color="black") p_s=plot(R*u+S*v, color="blue", legend_label="sum", legend_color="black") p_t=plot(target, color="black", legend_label="goal", legend_color="black") show(p_u + p_v + p_s + p_t) # the target was hit

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#51 # point (3, -7) W*vector([3,-7])

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(-5.61, 8.61)

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#51 check @interact def plotvectors(R=(-10,10, 0.1), S=(-10,10, 0.1), target=vector([3, -7])): u=R*vector([1,2]) v=S*vector([1,0.5]) p_u=plot(u, color="red", aspect_ratio=1.0, legend_label="u", legend_color="black") p_v=plot(v, color="green", legend_label="v", legend_color="black") p_s=plot(R*u+S*v, color="blue", legend_label="sum", legend_color="black") p_t=plot(target, color="black", legend_label="goal", legend_color="black") show(p_u + p_v + p_s + p_t) # the target was hit

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#52 T.inverse()

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[-0.3333333333333333 0.6666666666666666]
[ 1.3333333333333333 -0.6666666666666666]

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#53 # the result from number 49 and number 52 are the same, the W matrix and the inverse T matrix are equal to each other

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#54 M=matrix(RDF, [[7,-4],[4,-3]]) u=vector([1,2]) v=vector([1,.5]) M.eigenvectors_right() # the eigenvalues are 5 and -1

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

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#55 W*M*T # this is a diagonalized matrix and the eigenvalues are 5 and -1. It is a diagonal matrix because it has values that are very close to 0 on the nondiagonals and this occurs because of the rounding difference. Since this is a diagonalized matrix, this means that the eigenvalues are 5 and -1 as computed in question 54.

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[ -0.99 5.551115123125783e-17]
[-0.010000000000000675 5.0]

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