By Amy Harris

1
Using Sage in a BasicLinear Algebra Course

• By Amy Harris

2
Vector Addition

• Output from Sage

• Input into Sage

• xvector((0,2))
• yvector((3,4))
• xy

• (3, 6)

Vector Multiplication Dot Product

• Output from Sage

• Input into Sage

xy
8
3
Norm of a Vector

• Input into Sage

• Output from Sage

• norm(x)
• norm(y)

• 2
• 5

4
Matrix Addition

• Input into Sage

• Output from Sage

• Amatrix(RDF,3,2,3,-2)
• Bmatrix(RDF,1,2,3,-2)
• AB
• (Click on Evaluate)

• 4.0 4.0
• 6.0 -4.0

Matrix Multiplication

• Input into Sage

• Output from Sage

AB (Click on Evaluate)
9.0 2.0 -3.0 10.0
5
Solving a System of Linear Equations

• Input into Sage

• Output from Sage

• Amatrix(RDF,2,2,1,1, 1,-1)
• Bmatrix(RDF,2,1,4, -2)
• A.augment(B)
• (Click on Evaluate)
• A.solve_right(B)
• (Click on Evaluate)

1.0 1.0 4.0 1.0 -1.0 -2.0
1.0 3.0
6
Matrix A Additional Data

• Input into Sage

• Output from Sage

• A.determinant()
• A.trace()
• A.characteristic_polynomial()
• A.eigenvalues()
• A.eigenvectors_left()
• A.rank()
• A.inverse()
• A.nullity()

• -12.0
• 1.0
• 1.0x2 - 1.0x - 12.0
• 4.0, -3.0
• 0.948683298051 0.316227766017
• -0.4472135955 0.894427191
• 2
• 0.166666666667 0.166666666667
• 0.25 -0.25
• 0

7
Matrix Decomposition LU Factorization

• Output from Sage

• Input into Sage

• P,L,U A.LU()
• P
• (Click on Evaluate)
• L
• (Click on Evaluate)
• U
• (Click on Evaluate)

• 1.0 0.0
• 0.0 1.0
• 1.0 0.0
• 1.0 1.0
• 3.0 2.0
• 0.0 -4.0