Linear Algebra with Sagemath

See the following for a more thorough and likely better tutorial:

http://linear.ups.edu/html/fcla.html (Rob Beezer's book on Linear Algebra)
http://www.gregorybard.com/Sage.html (Greg Bard's book on Sagemath for undergraduates)

This worksheet is just more specialized for our 308 class

What's with the `QQ`?

In math, there are different number systems. We call them rings https://en.wikipedia.org/wiki/Ring_(mathematics). Linear algebra over different rings give you different answers. In this class, we'll use QQ for exact answer and RR for inexact ones.

What is Sage?

A computer algebra system built on top of Python.

Echelon Forms

Problem 1 http://kevinlui.org/m308exams/au17_midterm1sol.pdf

In [16]:

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

In [18]:

[ 1  2 -1  3]
[ 0  1 -1  2]
[ 0  0  1  1]

In [17]:

show(A)

In [19]:

B = A.rref()

In [20]:

show(B)

Solving Linear System

In [2]:

### Problem 1 <http://kevinlui.org/m308exams/au17_midterm1sol.pdf>

In [2]:

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

In [4]:

b = vector(QQ, [1, 1, 0])

In [9]:

show(A)

In [11]:

show(b)

In [12]:

A.solve_right(b)

(-1, 1, 0, 0)

In [6]:

A.right_kernel()

Vector space of degree 4 and dimension 1 over Rational Field
Basis matrix:
[   1 -3/2 -1/2  1/2]

The general solution is then $x=(-1,1,0,0)+s_1(1,-3/2,-1/2,1/2)$ .