midterm2.ipynb

Kernel: SageMath 8.0

Problem 1

B = matrix([[1,1,11],[-1,0,15],[1,2,2017]])

B^(-1)

[   -1/66 -133/132    1/132]
[ 508/495 1003/990  -13/990]
[  -1/990  -1/1980   1/1980]

y = vector([1,-2,0])

B \ y

(2, -1, 0)

Problem 4

A = matrix([[1,2,-1,-3],[2,4,0,-4],[3,6,-1,-7]])
A

[ 1  2 -1 -3]
[ 2  4  0 -4]
[ 3  6 -1 -7]

A.rref()

[ 1  2  0 -2]
[ 0  0  1  1]
[ 0  0  0  0]

Problem 5

A = matrix([[2,4,-1,-2],[-1,-3,-1,0],[1,1,2,2],[2,6,2,0]])
A

[ 2  4 -1 -2]
[-1 -3 -1  0]
[ 1  1  2  2]
[ 2  6  2  0]

B=A.rref()
B

[   1    0    0  1/2]
[   0    1    0 -1/2]
[   0    0    1    1]
[   0    0    0    0]

SB = A.columns()[:2]
TB = A.columns()[2:]

V = VectorSpace(QQ, 4)

S = V.subspace(SB)

T = V.subspace(TB)

S, T

(Vector space of degree 4 and dimension 2 over Rational Field
 Basis matrix:
 [ 1  0  1  0]
 [ 0  1  1 -2], Vector space of degree 4 and dimension 2 over Rational Field
 Basis matrix:
 [ 1  0 -1  0]
 [ 0  1 -1 -2])

S.intersection(T)

Vector space of degree 4 and dimension 1 over Rational Field
Basis matrix:
[ 1 -1  0  2]

SB[0]-SB[1]

(-2, 2, 0, -4)