# Lab 5

# Name: Sarah Ahmed
# 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 put each exercise in a new cell with an exercise number at the top.
# Use enough comments that you and the grader can understand your code.
# Label axes on all graphs.

#I changed the title to lab 5 because before it said lab 1!

#Lab 5: do #1-18

#1
v1=vector([3,10]) #create vector using matrix with values 3 and 10
v1 #show vector

#2
v=vector([-2, 7, 5]) #create vector with three-value matrix
type(v) #show what type of sage item the above vector is

#3
a=vector([4,2]) #define a as vector of matrix [4,2]
b=vector([-1,5]) #define b as vector of maatrix [-1,5]

a+b #add vectors
6*a #scale vector
4*a+3*b #add scaled vectors
-b #scale vector
-a+b #scale and add vectors
pi*a #scale vector

#4
v=vector([1,2,4]) #define v as vector of matrix [1,2,4]
plot(v, thickness=4) #plot vector with increased thickness

#5
a=vector([4,2]) #define a as vector of matrix [4,2]
b=vector([-1,5]) #define b as vector of matrix [-1,5]

#assign variables to the vector additions/scales from Exercise 3
v1=a+b
v2=6*a
v3=4*a+3*b
v4=-b
v5=-a+b
v6=pi*a

#plot the above defined vectors overlayed onto the original vectors (a and b)
plot(a, thickness=4, color="turquoise") + plot(b, thickness=4, color="teal") + plot(v1, color="green")
plot(a, thickness=4, color="turquoise") + plot(b, thickness=4, color="teal") + plot(v2, color="green")
plot(a, thickness=4, color="turquoise") + plot(b, thickness=4, color="teal") + plot(v3, color="green")
plot(a, thickness=4, color="turquoise") + plot(b, thickness=4, color="teal") + plot(v4, color="green")
plot(a, thickness=4, color="turquoise") + plot(b, thickness=4, color="teal") + plot(v5, color="green")
plot(a, thickness=4, color="turquoise") + plot(b, thickness=4, color="teal") + plot(v6, color="green")

#6
#write function that proves linearity given input of a function
#use variables in place of numbers
def lineartest(f): #define function name and single input
    var("a,b,c") #define a/b/c as variables
    if (f(a+b))==(f(a)+f(b)) and f(c*a)==c*f(a): #begin if loop and say that if conditions of linearity are met
        print("linear function") #print that the function is linear
    else: #end if loop and say that in all other cases
        print("nonlinear function") #print that the function is not linear

lineartest(3*x) #should be linear
lineartest(4+x) #should be nonlinear

#7
#test linearity-determining function on three different functions of one variable; one should be in the form k*x
lineartest(4*x) #should be linear
lineartest(x^2) #should be nonlinear
lineartest(tan(x)) #should be nonlinear

#8
def f(v): #define function that applies a vector to a linear function and exports the new vector GIVEN an input of a vector
    l1=[4*v[0]-2*v[1], 3*v[0]+v[1]] #assign l1 to the calculation of the new vector values given an input of old vector values
    v1=vector(l1) #assign new vector to variable v1
    return v1 #return new vector

v=vector([a,b]) #assign variable to vector from matrix [a,b]
f(v) #run variable through above function. expected result: (4a-2b, 3a+b)

#9
def linearv2(v): #define another function that does the same as f(v) but applies a different linear function
    l1=[5*v[0]-2*v[1], 1*v[0]-3*v[1]] #new vector should be these calculations done to the corresponding values of the last vector
    v1=vector(l1) #assign variable to the new vector
    return v1 #return new vector

v=vector([a,b]) #assign variable to the vector of matrix [a,b]
linearv2(v) #run variable through above function. expected result: (5a-2b, a-3b)

#assign variables to three different vectors
vec1=vector([3,2])
vec2=vector([7,1])
vec3=vector([9,5])

#run vectors through function
vec1_1=linearv2(vec1)
vec2_1=linearv2(vec2)
vec3_1=linearv2(vec3)

#plot original vectors overlayed with corresponding new vectors
plot(vec1, color="turquoise") + plot(vec1_1, color="teal")
plot(vec2, color="turquoise") + plot(vec2_1, color="teal")
plot(vec3, color="turquoise") + plot(vec3_1, color="teal")

#10
def linearv3(v): #define another function that does the same as f(v) and linearv2(v) but with a different linear function
    l1=[4*v[0]+5*v[1]-7*v[2], 2*v[0]-v[1]+6*v[2], 3*v[0]+5*v[1]-3*v[2]] #assign variable to the calculated new values using old vector values. this one gives us three values!
    v1=vector(l1) #assign variable to the new vector
    return v1 #show new vector

vec1=vector([2,2,2]) #assign variable to the vector of matrix [2,2,2]
linearv3(vec1) #run variable through function. expected result (4,14,10)

#11
def blackbearvec(v): #define function; input vector is in the from (j(t), a(t)) where j(t) is the number of juvenile blackbears at time=t and a(t) is the number of adult blackbears at time=t
    var("t, t_next") #t_next is the variable used to
    j(t_next)=.57*v[0]+.5205*v[1] #calculation for value of j at time=t+1
    a(t_next)=.33*v[0]+.917*v[1] #calculation for value of a at time=t+1
    l1=[j(t_next), a(t_next)] #assign list to new j and a values
    v1=vector(l1) #assign variable to new vector
    return v1 #show new vector

v=vector([1,1]) #assign variable to vector of matrix [1,1] AKA use initial conditions j=1 and a=1
blackbearvec(v) #run through function to get values at t+1

#12
v=vector([13,20]) #assign variable to vector (13,20)
v2=blackbearvec(v) #run variable through function
plot(v, color="turquoise") + plot(v2, color="teal", aspect_ratio=1, axes_labels=["juveniles", "adults"]) #plot original vector overlayed with new vector
v2
"At time=t+1 there will be 17 juveniles and 22 adults."

#13
vec1=vector([1,0]) #assign variable to vector (1,0)
vec2=vector([0,1]) #assign variable to vector (0,1)

#run vectors through function
vec1_1=blackbearvec(vec1)
vec2_1=blackbearvec(vec2)

#plot orignal and new vectors overlayed upon one another
plot(vec1, color="turquoise") + plot(vec1_1, color="teal", aspect_ratio=1, axes_labels=["juveniles", "adults"])
plot(vec2, color="mediumseagreen", aspect_ratio=1) + plot(vec2_1, color="green", aspect_ratio=1, axes_labels=["juveniles", "adults"])

#plot all vectors together
plot(vec1, color="turquoise") + plot(vec2, color="mediumseagreen", aspect_ratio=1) + plot(vec1_1, color="teal", aspect_ratio=1) + plot(vec2_1, color="green", aspect_ratio=1, axes_labels=["juveniles", "adults"])

#14
vec1=vector([1,0]) #assign variable to vector (1,0)
vec2=vector([0,1]) #assign variable to vector (0,1)
vec1_1=13*blackbearvec(vec1) #assign variable to result of scaling the vector resulting from inputting vec1 into blackbear function
vec2_1=20*blackbearvec(vec2) #same as above but using vec2
plot(vec1_1, color="teal", aspect_ratio=1) + plot(vec2_1, color="green", aspect_ratio=1) + plot(vec1_1+vec2_1, color="mediumseagreen", aspect_ratio=1, axes_labels=["juveniles", "adults"]) #plot the two product vectors individually and added together

v=vector([13,20]) #assign variable to vector (13,20)
v2=blackbearvec(v) #run vector through blackbear function
plot(v, color="turquoise") + plot(v2, color="teal", aspect_ratio=1, axes_labels=["juveniles", "adults"]) #plot original and new vectors -- #compare to above to this (vector from 12)

"They are the same final vector! (compare light green vector in top to teal vector in bottom)"

#15
#repeat 14 for two other sets of IC -- same descriptions
vec1=vector([1,0])
vec2=vector([0,1])
vec1_1=5*blackbearvec(vec1)
vec2_1=50*blackbearvec(vec2)
plot(vec1_1, color="teal", aspect_ratio=1) + plot(vec2_1, color="green", aspect_ratio=1) + plot(vec1_1+vec2_1, color="mediumseagreen", aspect_ratio=1, axes_labels=["juveniles", "adults"])

v=vector([5,50])
v2=blackbearvec(v)
plot(v, color="turquoise") + plot(v2, color="teal", aspect_ratio=1, axes_labels=["juveniles", "adults"]) #compare to vector from 12

vec1=vector([1,0])
vec2=vector([0,1])
vec1_1=14*blackbearvec(vec1)
vec2_1=12*blackbearvec(vec2)
plot(vec1_1, color="red", aspect_ratio=1) + plot(vec2_1, color="gold", aspect_ratio=1) + plot(vec1_1+vec2_1, color="orange", aspect_ratio=1, axes_labels=["juveniles", "adults"])

v=vector([14,12])
v2=blackbearvec(v)
plot(v, color="pink") + plot(v2, color="orange", aspect_ratio=1, axes_labels=["juveniles", "adults"]) #compare to vector from 12

#16
#first linear function
def linear1(v): #define function; input vector is in the from (a(t), b(t)) where a(t) is the a value at time=t and b(t) is the b value at time=t
    var("t, t_next") #t_next is the variable used to represent the values for time=t+1
    j(t_next)=6*v[0]+.13*v[1] #equation for value of j at time=t+1
    a(t_next)=14*v[0]-.65*v[1] #equation for value of a at time=t+1
    l1=[j(t_next), a(t_next)] #define variable for list of t+1 values of j and a
    v1=vector(l1) #make list into vector and assign that vector to a variable
    return v1 #return vector

v=vector([3,4]) #create vector
linear1(v) #testing function - run vector through function

#apply to vector
vec1=vector([1,0]) #create first vector
vec2=vector([0,1]) #create second vector
vec1_1=5*linear1(vec1) #scale the vector produced from running first vector through function
vec2_1=7*linear1(vec2) #same as above but by running second vector through function
plot(vec1_1, color="teal", aspect_ratio=1) + plot(vec2_1, color="green", aspect_ratio=1) + plot(vec1_1+vec2_1, color="mediumseagreen", aspect_ratio=1) #plot original vectors plus the sum of the new vectors, all overlayed

v=vector([5,7]) #create vector
v2=linear1(v) #run vector through function
plot(v, color="turquoise") + plot(v2, color="teal", aspect_ratio=1) #plot original and new vectors

#second linear function -- same as above, but with different functions to define a and j and time=t+1
def linear2(v): #define function; input vector is in the from (a(t), b(t)) where a(t) is the a value at time=t and b(t) is the b value at time=t
    var("t, t_next") #t_next is the variable used to represent values for time=t+1
    j(t_next)=.5*v[0]+.01*v[1] #define value of j at time=t+1
    a(t_next)=-.3*v[0]+.7*v[1] #define value of a at time=t+1
    l1=[j(t_next), a(t_next)] #create list of values of a and j at time=t+1
    v1=vector(l1) #turn list into vector
    return v1 #return vector

v=vector([3,4]) #create vector
linear2(v) #testing function - run vector through function

#apply to vector
vec1=vector([1,0]) #same as above
vec2=vector([0,1])
vec1_1=14*linear2(vec1)
vec2_1=12*linear2(vec2)
plot(vec1_1, color="red", aspect_ratio=1) + plot(vec2_1, color="gold", aspect_ratio=1) + plot(vec1_1+vec2_1, color="orange", aspect_ratio=1)

v=vector([14,12]) #same as above
v2=linear2(v)
plot(v, color="pink") + plot(v2, color="orange", aspect_ratio=1)

#17
vec1=vector([1,0]) #create first vector
vec2=vector([0,1]) #create second vector
bb01=blackbearvec(vec1) #run first vector through function
bb10=blackbearvec(vec2) #run second vector through function
bb01 #return new vector corresponding with vector 1
bb10 #return new vector corresponding with vector 2

#construct column matrix
ex1=vector([3,5]) #first column vector
ex2=vector([4,6]) #second column vector
column_matrix([ex1,ex2]) #matrix with two columns put together

#18
column_matrix([bb01, bb10]) #construct column matrix out of the new vectors fom #17