# Alex Moberg, Introduction to Linear Algebra in Sage
# (1) Solve the following systems of linear equations by putting them in row reduced echelon form: x-2y+z=0, 2xy-8z=8, -4x+5y+9z=-9
A=matrix(3,4,[1,-2,1,0,0,2,-8,8,-4,5,9,-9])
print A.rref()

# (2) Let A equal the following matrix and let B equal the vector <6, 15, 24>.  Use Sage to find all solutions to the equation Ax=B.
A=matrix(3,3,[1,2,3,4,5,6,7,8,9]);show(A)
B=vector([6,15,24])
A\B
C=matrix(3,4,[1,2,3,6,4,5,6,15,7,8,9,24]);show(C.rref())

# (3) Let A and B equal the following matrices.  Compute A+3B, B-2A, AB, and B-3I.
A=matrix(2,2,[1,2,-2,1]); show(A); B=matrix(2,2,[3,5,-1,4]); show(B)
A+3*B
B-2*A
A*B
I=identity_matrix(2)
B-3*I

# (4) Let U equal the vector <1,1,0> and V equal the vector <2,0,-1>.  Compute the dot product of U*V, the cross product of U x V and the magnitude of V.
U=vector([1,1,0])
V=vector([2,0,-1])
U.dot_product(V)
U.cross_product(V)
norm(V)

# (5) Let V equal the vector <-2,4,2,2> and W equal the vector <0,3,5,2>.  Combine vector operations to compute the angle theta between V and W.
V=vector([-2,4,2,2])
W=vector([0,3,5,2])
dotProductVW=V.dot_product(W)
cosOftheta=dotProductVW/ (norm(V)*norm(W))
theta=arccos(cosOftheta)
print(n(theta))