CoCalc Public FilesFall2020_Lecture_Books / cs410_2020_lec01n01.ipynb
Authors: Ross Beveridge, V K, Yongxin Liu
Views : 133
Description: First CS410 SageMath Notebook for Fall 2020
Compute Environment: Ubuntu 20.04 (Default)

### First 410 Notebook Illustrating SageMath on CoCalc

This is a tiny step down the path we will take this Fall using SageMath through Jupyter - and hosted by CoCalc - to illustrate key concepts during the course of CS410. Today, the very basics of formulas and linear algebra.

Ross Beveridge, August 25, 2020

In [33]:
%display latex
latex.matrix_delimiters(left='|', right='|')
latex.vector_delimiters(left='[', right=']')


To begin, you can designate variables that will be treated as symbolic as opposed to values.

In [34]:
var('a','b','c','d','x','y')

$\left(a, b, c, d, x, y\right)$

Next you can create vectors and matrices which are symbolic (not numeric - yet)

In [39]:
mm = matrix([[a,b],[c,d]])
uu = vector([x,y])
mm,uu

$\left(\left|\begin{array}{rr} a & b \\ c & d \end{array}\right|, \left[x,\,y\right]\right)$

But notice all we have done so far is to implicitly create a sequence of the two objects, the matrix and the vector, and print them back in the basic Python/Jupyter/SageMath Read-Eval-Pring Loop. What if we want to create better formed equations?

In [40]:
pretty_print("m = ", mm, "  and", "  u = ", uu)

$\verb|m|\phantom{\verb!x!}\verb|=| \left|\begin{array}{rr} a & b \\ c & d \end{array}\right| \phantom{\verb!xx!}\verb|and| \phantom{\verb!xx!}\verb|u|\phantom{\verb!x!}\verb|=| \left[x,\,y\right]$

Next complication is that a matrix times a vector becomes somewhat an issue deep in the particulars of any given language. Conceptually, it may at times be simplest to realize that a vector is - when doing matrix multiplication - just a one dimensional matrix. So below we arrive at the simple case of a 2x2 matrix times a 2x1 vector/matrix

In [37]:
um = matrix(uu).transpose()
pretty_print(LatexExpr("M = "), mm, ", ", LatexExpr("U = "), um)

$M = \left|\begin{array}{rr} a & b \\ c & d \end{array}\right| \verb|,| U = \left|\begin{array}{r} x \\ y \end{array}\right|$
In [38]:
vm = mm * um
pretty_print(vm, " = ", mm, "*", um)

$\left|\begin{array}{r} a x + b y \\ c x + d y \end{array}\right| \phantom{\verb!x!}\verb|=| \left|\begin{array}{rr} a & b \\ c & d \end{array}\right| \verb|*| \left|\begin{array}{r} x \\ y \end{array}\right|$

And while it may not seem like much, notice that SageMath carried out the symbolic computation of multiplying a matrix and a column vector. Put simply, anyone spending much of their life working with linear algebra in contexts such as computer graphics, should master a tool such as SageMath (Maple, Mathematic, ..). The reason is that true understanding comes from personal experience working back and forth between multiple ways of conceptualizing a problem, and hence any tool that does the routine part for you is helpful.