Understanding the Consistency of Homogeneous Linear Systems in Linear Algebra

A linear system whose equations are all homogeneous must be consistent.

In order to understand why a linear system whose equations are all homogeneous must be consistent, let’s start by defining what a homogeneous system and a consistent system mean in the context of linear algebra

In order to understand why a linear system whose equations are all homogeneous must be consistent, let’s start by defining what a homogeneous system and a consistent system mean in the context of linear algebra.

1. Homogeneous system: A system of linear equations is said to be homogeneous if the constant terms in each equation are all zero. In other words, a system is homogeneous if it can be written in the form:

a₁x₁ + a₂x₂ + … + aₙxₙ = 0
b₁x₁ + b₂x₂ + … + bₙxₙ = 0
…
c₁x₁ + c₂x₂ + … + cₙxₙ = 0

Where a₁, a₂, …, aₙ, b₁, b₂, …, bₙ, …, c₁, c₂, …, cₙ are the coefficients of the variables x₁, x₂, …, xₙ respectively.

2. Consistent system: A system of linear equations is said to be consistent if there exists at least one solution, or in other words, if there is a set of values for the variables that satisfies all the equations simultaneously.

Now, let’s prove why a linear system whose equations are all homogeneous (i.e., have zero constant terms) must be consistent.

For simplicity, let’s consider a system of two linear equations in two variables:

a₁x + b₁y = 0
a₂x + b₂y = 0

We can rewrite this system in matrix form as:

AX = 0

Where:
A = [a₁ b₁]
[a₂ b₂]

X = [x]
[y]

The homogeneous system AX = 0 always has at least one solution, which is the trivial solution X = [0, 0]. This is because replacing the variables x and y with zero in the equations satisfies them.

To see why there cannot be any other solutions for a homogeneous system, let’s assume there exists a non-trivial solution X ≠ [0, 0]. This would mean that at least one of the variables x or y has a non-zero value. But since the system is homogeneous, plugging this non-zero value into the equations would violate the property of being homogeneous, as we would have a non-zero constant term on the right-hand side.

Hence, the assumption of having a non-trivial solution is contradictory, and therefore a homogeneous linear system always has the trivial solution X = [0, 0]. Consequently, a homogeneous system is consistent since it has at least one solution.

The same reasoning can be extended to linear systems with more equations and variables, confirming that any linear system whose equations are all homogeneous must be consistent.

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