Polynomials and linear algebra

Overview

Teaching: 30 min
Exercises: 0 min

Questions

• …

Objectives

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Next we demonstrate indeterminates and polynomials. This is an indeteminate x. It is defined automatically when SageMath starts (however, it is not read-only and may be easily overwritten):

x

x

One can construct polynomials and find their roots as follows:

f=x^2-x-1

f

x^2 - x - 1

solve([f==0],x)

[x == -1/2*sqrt(5) + 1/2, x == 1/2*sqrt(5) + 1/2]

Convert this these to a floating point variable:

float(1/2*sqrt(5) + 1/2)

1.618033988749895

Other symbolic variables are not defined. For example, if we try to define v to be the vector with symbolic coordinates x, y, z, we will have an error, because only x is defined by default when you launch Sage!

v = vector([x, y, z])

---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-6-41c893d7dada> in <module>()
----> 1 v = vector([x, y, z])

NameError: name 'y' is not defined

To fix this, we need to define y (and z) as symbolic variables, using SR.var (where “SR” stands for “symbolic ring”):

x, y, z = SR.var("x y z")

v = vector([x, y, z])
v

(x, y, z)

Such vectors can be operated with in the same way like vectors with numeric coordinates. To demonstrate this, we want to create an additive inverse of a 3x3 indentity matrix over integers. One could certainly enter it with matrix([[-1,0,0], [0,-1,0], [0,0,-1]]), but actually SageMath has a function identity_matrix that can be used instead. Below we demonstrate how to use SageMath help system to get its documentation:

?identity_matrix

Hence, we will define A using this special function.

A = -identity_matrix(3)
A

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

Now we multiply this matrix and symbolic vector using *

A * v

(-x, -y, -z)

We can substitute values of variables as follows:

v.subs(x=1, y=0, z=3)

(1, 0, 3)

A * v.subs(x=1, y=0, z=3)

(-1, 0, -3)

A * v

(-x, -y, -z)

_.subs(x=1, y=0, z=3)

(-1, 0, -3)

In the cell above, the underscore _ refers to the result of the last executed command, that is of A*v.

Key Points