Lab 6.ipynb

Kernel: SageMath 9.2

# Name: 
# 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.

#43a
@interact
def vectors(R=(-10,10,0.01),S=(-10,10,0.01), target=vector([2,3])):
    u= R*vector([1,2])
    v= S*vector([1,0.5])
    pu= plot(u,color="dodgerblue", legend_label="R * u",legend_color="black")
    pv= plot(v,color="lime", legend_label= "S * v",legend_color="black")
    padd= plot(u+v, color="orchid",legend_label="R * u + S * v",legend_color="black")
    ptarget= point(target,size=40,color="turquoise",legend_label="[2,3]",legend_color="black")
    show(pu+pv+padd+ptarget,legend_loc='upper left')

#R= 1.35 and S=0.65 

#43b
@interact
def vectors(R=(-10,10,0.01),S=(-10,10,0.01), target=vector([-5,1.2])):
    u= R*vector([1,2])
    v= S*vector([1,0.5])
    pu= plot(u,color="dodgerblue", legend_label="R * u",legend_color="black")
    pv= plot(v,color="lime", legend_label= "S * v",legend_color="black")
    padd= plot(u+v, color="orchid",legend_label="R * u + S * v",legend_color="black")
    ptarget= point(target,size=40,color="turquoise",legend_label="[-5,1.2]",legend_color="black")
    show(pu+pv+padd+ptarget,legend_loc='lower left')

#R= 2.46 and S= -7.46

#43c
@interact
def vectors(R=(-10,10,0.01),S=(-10,10,0.01), target=vector([-4,-8])):
    u= R*vector([1,2])
    v= S*vector([1,0.5])
    pu= plot(u,color="dodgerblue", legend_label="R * u",legend_color="black")
    pv= plot(v,color="lime", legend_label= "S * v",legend_color="black")
    padd= plot(u+v, color="orchid",legend_label="R * u + S * v",legend_color="black")
    ptarget= point(target,size=40,color="turquoise",legend_label="[-4,-8]",legend_color="black")
    show(pu+pv+padd+ptarget,legend_loc='upper left')

# R=-4 and S=0

#43d
@interact
def vectors(R=(-10,10,0.01),S=(-10,10,0.01), target=vector([3,-2.4])):
    u= R*vector([1,2])
    v= S*vector([1,0.5])
    pu= plot(u,color="dodgerblue", legend_label="R * u",legend_color="black")
    pv= plot(v,color="lime", legend_label= "S * v",legend_color="black")
    padd= plot(u+v, color="orchid",legend_label="R * u + S * v",legend_color="black")
    ptarget= point(target,size=40,color="turquoise",legend_label="[3,-2.4]",legend_color="black")
    show(pu+pv+padd+ptarget,legend_loc='upper left')

# R= -2.6 and S= 5.62

#44a
@interact
def vectors(R=(-10,10,0.01),S=(-10,10,0.01), target=vector([2,3])):
    u= R*vector([2,3])
    v= S*vector([1,0.4])
    pu= plot(u,color="dodgerblue", legend_label="R * u",legend_color="black")
    pv= plot(v,color="lime", legend_label= "S * v",legend_color="black")
    padd= plot(u+v, color="orchid",legend_label="R * u + S * v",legend_color="black")
    ptarget= point(target,size=40,color="turquoise",legend_label="[2,3]",legend_color="black")
    show(pu+pv+padd+ptarget,legend_loc='upper left')
# R=1 and S=0

#44b
@interact
def vectors(R=(-10,10,0.01),S=(-10,10,0.01), target=vector([-5,1.2])):
    u= R*vector([2,3])
    v= S*vector([1,0.4])
    pu= plot(u,color="dodgerblue", legend_label="R * u",legend_color="black")
    pv= plot(v,color="lime", legend_label= "S * v",legend_color="black")
    padd= plot(u+v, color="orchid",legend_label="R * u + S * v",legend_color="black")
    ptarget= point(target,size=40,color="turquoise",legend_label="[-5,1.2]",legend_color="black")
    show(pu+pv+padd+ptarget,legend_loc='lower left')
# R =1.48 and S = -8

#44c
@interact
def vectors(R=(-10,10,0.01),S=(-10,10,0.01), target=vector([-4,-8])):
    u= R*vector([2,3])
    v= S*vector([1,0.4])
    pu= plot(u,color="dodgerblue", legend_label="R * u",legend_color="black")
    pv= plot(v,color="lime", legend_label= "S * v",legend_color="black")
    padd= plot(u+v, color="orchid",legend_label="R * u + S * v",legend_color="black")
    ptarget= point(target,size=40,color="turquoise",legend_label="[-4,-8]",legend_color="black")
    show(pu+pv+padd+ptarget,legend_loc='lower left')
# R = -2.9 and S = 1.78

#44d
@interact
def vectors(R=(-10,10,0.01),S=(-10,10,0.01), target=vector([3,-2.4])):
    u= R*vector([2,3])
    v= S*vector([1,0.4])
    pu= plot(u,color="dodgerblue", legend_label="R * u",legend_color="black")
    pv= plot(v,color="lime", legend_label= "S * v",legend_color="black")
    padd= plot(u+v, color="orchid",legend_label="R * u + S * v",legend_color="black")
    ptarget= point(target,size=40,color="turquoise",legend_label="[3,-2.4]",legend_color="black")
    show(pu+pv+padd+ptarget,legend_loc='lower left')
# R =1.63 and S = 6.27

#45a
@interact
def vectors(R=(-10,10,0.01),S=(-10,10,0.01), target=vector([2,3])):
    u= R*vector([1,2])
    v= S*u
    pu= plot(u,color="dodgerblue", legend_label="R * u",legend_color="black")
    pv= plot(v,color="lime", legend_label= "S * u",legend_color="black")
    padd= plot(u+v, color="orchid",legend_label="R * u + S * u",legend_color="black")
    ptarget= point(target,size=40,color="turquoise",legend_label="[2,3]",legend_color="black")
    show(pu+pv+padd+ptarget,legend_loc='upper left')
#Cannot reach the target point when u is a scalar multiple of v

#45b
@interact
def vectors(R=(-10,10,0.01),S=(-10,10,0.01), target=vector([-5,1.2])):
    u= R*vector([1,2])
    v= S*u
    pu= plot(u,color="dodgerblue", legend_label="R * u",legend_color="black")
    pv= plot(v,color="lime", legend_label= "S * u",legend_color="black")
    padd= plot(u+v, color="orchid",legend_label="R * u + S * u",legend_color="black")
    ptarget= point(target,size=40,color="turquoise",legend_label="[-5,1.2]",legend_color="black")
    show(pu+pv+padd+ptarget,legend_loc='lower left')
#Cannot reach the target point when u is a scalar multiple of v

#45c
@interact
def vectors(R=(-10,10,0.01),S=(-10,10,0.01), target=vector([-4,-8])):
    u= R*vector([1,2])
    v= S*u
    pu= plot(u,color="dodgerblue", legend_label="R * u",legend_color="black")
    pv= plot(v,color="lime", legend_label= "S * u",legend_color="black")
    padd= plot(u+v, color="orchid",legend_label="R * u + S * u",legend_color="black")
    ptarget= point(target,size=40,color="turquoise",legend_label="[-4,-8]",legend_color="black")
    show(pu+pv+padd+ptarget,legend_loc='upper left')
#We can reach the target point when u is a scalar multiple of v at R=4 and S=-2

#45d
@interact
def vectors(R=(-10,10,0.01),S=(-10,10,0.01), target=vector([3,-2.4])):
    u= R*vector([1,2])
    v= S*u
    pu= plot(u,color="dodgerblue", legend_label="R * u",legend_color="black")
    pv= plot(v,color="lime", legend_label= "S * u",legend_color="black")
    padd= plot(u+v, color="orchid",legend_label="R * u + S * u",legend_color="black")
    ptarget= point(target,size=40,color="turquoise",legend_label="[3,-2.4]",legend_color="black")
    show(pu+pv+padd+ptarget,legend_loc='upper left')
#Cannot reach the target point when u is a scalar multiple of v

#46
twobytwo= matrix([[1,1],[2,1/2]])
show(twobytwo)

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 1 & 1 \\ 2 & \frac{1}{2} \end{array}\right)$$

#47
show("Product:",twobytwo*vector([1.35,0.65]))
#The product is very close to the target point we were trying to reach which was [2,3]

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\verb|Product:| \left(2.00000000000000,\,3.02500000000000\right)$$

#48
Matrix1=twobytwo*vector([2.46,-7.46]) # Target: [-5,1.2]
Matrix2=twobytwo*vector([-4,0]) #Target: [-4,-8]
Matrix3=twobytwo*vector([-2.6,5.62]) #Target: [3,-2.4]
show("Product 1:",Matrix1)
show("Product 2:",Matrix2)
show("Product 3:",Matrix3)

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\verb|Product|\phantom{\verb!x!}\verb|1:| \left(-5.00000000000000,\,1.19000000000000\right)$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\verb|Product|\phantom{\verb!x!}\verb|2:| \left(-4,\,-8\right)$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\verb|Product|\phantom{\verb!x!}\verb|3:| \left(3.02000000000000,\,-2.39000000000000\right)$$

#49
M=matrix([[-1/3,2/3],[4/3,-2/3]])
show(M)

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} -\frac{1}{3} & \frac{2}{3} \\ \frac{4}{3} & -\frac{2}{3} \end{array}\right)$$

#50a
beep=vector([1,-6])
show("Coordinates of [R,S]:",M*beep)

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\verb|Coordinates|\phantom{\verb!x!}\verb|of|\phantom{\verb!x!}\verb|[R,S]:| \left(-\frac{13}{3},\,\frac{16}{3}\right)$$

#50b
@interact
def vecfunc(R=(-10,10,0.05),S=(-10,10,0.05),target=vector([1,-6])):
    u= R*vector([1,2])
    v= S*vector([1,1/2])
    P1= plot(u,legend_label="u",legend_color="black",color="lime")
    P2= plot(v,legend_label="v",legend_color="black")
    P3= plot(u+v,legend_label="u + v",legend_color="black",color="gold")
    point1= point(u+v,size=50,color="turquoise",legend_label= "u + v point",legend_color="black")
    point2= point(target,size=50,color="orchid",legend_label="Target: [1,-6]",legend_color="black")
    show(P1+P2+P3+point1+point2)
# R=-4.33 and S=5.33 are the coordinates that hit the target

#51a
beep=vector([-4,-8])
show("Coordinates of [R,S]:",M*beep)

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\verb|Coordinates|\phantom{\verb!x!}\verb|of|\phantom{\verb!x!}\verb|[R,S]:| \left(-4,\,0\right)$$

#51b
@interact
def vecfunc(R=(-10,10,0.05),S=(-10,10,0.05),target=vector([-4,-8])):
    u= R*vector([1,2])
    v= S*vector([1,1/2])
    P1= plot(u,legend_label="u",legend_color="black",color="lime")
    P2= plot(v,legend_label="v",legend_color="black")
    P3= plot(u+v,legend_label="u + v",legend_color="black",color="orchid")
    point1= point(u+v,size=30,color="turquoise",legend_label= "u + v point",legend_color="black")
    point2= point(target,size=50,color="gold",legend_label="Target: [1,-6]",legend_color="black")
    show(P1+P2+P3+point1+point2)
# R=-4 and S=0 are the coordinates that hit the target

#52
show(twobytwo.inverse())

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} -\frac{1}{3} & \frac{2}{3} \\ \frac{4}{3} & -\frac{2}{3} \end{array}\right)$$

#53
#The inverse matrix of T  (or in my case 2 x 2 matrix) is the same as the M matrix

#54
Mtrix= matrix([[7,-4],[4,-3]])
k=Mtrix.eigenvectors_right()
show(Mtrix)
show(k)
#The eigenvalue of (1,2) is -1 and the eigenvalue of (1,1/2) is 5

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 7 & -4 \\ 4 & -3 \end{array}\right)$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\left[\left(5, \left[\left(1,\,\frac{1}{2}\right)\right], 1\right), \left(-1, \left[\left(1,\,2\right)\right], 1\right)\right]$$

#55
First=matrix([[1,1],[2,1/2]])
Second= matrix([[7,-4],[4,-3]])
Third= First.inverse()
kek=Third*Second*First
show("Diagonal Matrix of 54 with Eigenvalues:",kek)
#The resulting matrix is the diagonal matrix of 54 with the eigenvalues of -1 and 5

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\verb|Diagonal|\phantom{\verb!x!}\verb|Matrix|\phantom{\verb!x!}\verb|of|\phantom{\verb!x!}\verb|54|\phantom{\verb!x!}\verb|with|\phantom{\verb!x!}\verb|Eigenvalues:| \left(\begin{array}{rr} -1 & 0 \\ 0 & 5 \end{array}\right)$$