Homogenous

Homogenous#

Homogeneous Linear Systems#

Let’s break down homogeneous systems of equations and their connection to nullspace, using SageMath examples to illustrate.

Definitions:

Homogeneous Systems of Equations

A homogeneous system of equations is simply a system of linear equations where the constant term (the value on the right-hand side of the equation) is zero for every equation. This gives us a general form like this:

\[\begin{split} \begin{align*} a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n &= 0 \\ a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n &= 0 \\ &\vdots\\ a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n &= 0 \end{align*} \end{split}\]

Here’s a key property:

Trivial Solution

Every homogeneous system always has at least one solution: the trivial solution where all the variables (x1, x2, …, xn) are equal to zero.

SageMath Example

Let’s consider the following homogeneous system:

2x + 3y - z = 0
x - 4y + 2z = 0

In SageMath, we represent this as a matrix equation: Ax = 0 where A is the coefficient matrix, x is the vector of variables, and 0 is the zero vector.

A = matrix([[2, 3, -1], [1, -4, 2]])
show(A)
\(\displaystyle \left(\begin{array}{rrr} 2 & 3 & -1 \\ 1 & -4 & 2 \end{array}\right)\)
b = vector([0, 0])  # Zero vector for a homogeneous system
show(b)
\(\displaystyle \left(0,\,0\right)\)

Nullspace (Kernel)#

In this section, we briefly cover nullspace. It will be covered in more detail in a later lesson.

Definition: nullspace

The nullspace (also called the kernel) of a matrix \(A\) is the set of all vectors \(x\) that satisfy the equation \(Ax = 0\).

In other words, it’s the set of all solutions to the homogeneous system of equations represented by matrix \(A\).

Finding the Nullspace in SageMath

SageMath makes it easy to calculate the nullspace:

nullspace = A.right_kernel()  # Find the right kernel (nullspace)
print(nullspace)

Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:
[  2  -5 -11]

Interpretation

The output [2, -5, 11] tells us that any scalar multiple of this vector is a solution to our homogeneous system.

For example:

  • [2, -5, 11] is a solution

  • [4, -10, 22] is a solution

  • [-2, 5, -11] is a solution

…and so on.

Summary#

  • Homogeneous systems have all zero constant terms.

  • They always have at least the trivial solution (all variables zero).

  • The nullspace of the coefficient matrix captures all solutions to the homogeneous system.

  • The nullspace reveals if there are non-trivial solutions and how those solutions are structured.

By understanding homogeneous systems and their solutions, we gain deeper insights into the nature of linear transformations and vector spaces, which are foundational concepts in both theoretical and applied mathematics.

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