Download Linear Equations in 3D Space

Experimental Mathematics
25 August 2011
Linear Equations in 2D Space
Recall: A line in the x-y-plane can be represented
algebraically by an equation of the form
ax + by = c
where a and b are not both zero.
Sage example for a=2, b=3, c=5, plotted over
-10≤x≤10, -10≤y≤10:
var('x,y')
p=implicit_plot(2*x+3*y==5, (x,-10,10),(y,-10,10))
show(p)
Problem 1
Open a new Sage worksheet and do today’s
problems within this worksheet.
Find the intersection point of the lines determined by
4x+y=10 and 3x-4y=17 by a Sage plot.
Hint: Two plots p,q can be combined by show(p+q)
Linear Equations in 3D Space
Recall: A linear equation ax + by + cz = d, where a,
b, c, d are constants and not all a, b, c are zero,
represents a plane in the three dimensional space.
Sage example for a=2, b=3, c=5, d=-2, plotted over
-10≤x≤10, -10≤y≤10, -10≤z≤10, in red color:
var(‘x,y,z’)
p=implicit_plot3d(2*x+3*y+5*z==-2, (x,-10,10),(y,-10,10),(z,-10,10),color=‘red’)
show(p)
Problem 2
Combine the three planes determined by
2x+3y+5z=10
x-y+z=1
x+y-2z=0
in one plot. Plot each plane in a different color.
Approximately identify the point where all three
planes intersect.
Problem 3
The system
x+y+z=-1
x-y+z=1
x+z=2
has no solution. Combine these three planes in one
plot (with different colors) and find the “geometric
reason” why there is no solution.
Vector and Matrix Operations in Sage
v = vector([1,2])
w = vector([1,3])
A = matrix([[1,2],[2,3]])
B = matrix([[1,0,0],[1,1,2]])
A^3
A*B
A*v
10*B
10*v
def f(i,j):
return i*j
C = matrix(10,10,f)
#2D vector
#2D vector
# 2x2 matrix
# 2x3 matrix
# matrix power
# matrix product
# matrix-vector product
# scalar multiplication
# scalar multiplication
# C is 10x10 matrix with
# C[i,j] = f(i,j)
Problem 4
a) Try out matrix and vector computations as on the
last slide.
Note: vectors are different from lists and matrices are different from
lists of lists. E.g. the append function does not work for vectors.
b) Create the following matrix in Sage (where x is a
symbolic variable):
 x 0 1


A = 0 1 1
 0 0 2


2
3
4
Compute A , A , A ,....
Can you guess what An is in general?
Solving Linear Systems
#Example:
b = vector([1,2,2])
A = matrix([[1,2,-1],[2,3,1],[1,-1,0]])
x=A.solve_right(b)
# solves Ax=b
A*x
# check
Problem 5
Use the solve_right function to solve the system
from Problem 2:
2x+3y+5z=10
x-y+z=1
x+y-2z=0
Polynomial Interpolation
Example
Find a polynomial of degree 2 through
(0,4),(1,5),(2,5):
# solve the linear system:
#Compute polynomial and test:
x=vector([0,1,2])
y=vector([4,5,5])
n=2
def f(i,j):
return x[i]^(n-j)
A = matrix(3,3,f)
coeffs = A.solve_right(y)
p(x) = 0
for i in range(3):
p+=coeffs[i]*x^(n-i)
print p
#test:
p(0); p(1); p(2)
Problem 6
a) Write a Python function Interpol(L) which
returns a polynomial of degree n which
passes through the n+1 points in the list L.
For example, if L=[[0,0],[1,1],[2,4]], the
2
x
function should return
b) Apply your function to the example on the
last slide and use it to find a polynomial of
degree 5 through
(0,4),(1,5),(2,5),(3,100),(4,-10),(5,-5).