Understanding the Vector Space of Homogeneous Matrix Equations

The solution set of a homogeneous matrix equation (Ax 0) is indeed a vector space. This article will explore why this statement is true and delve into the fundamental concepts of vector spaces in the context of linear algebra.

Definition of Homogeneous System

A homogeneous system of linear equations is represented by the matrix equation (Ax 0), where (A) is an (m times n) matrix and (x) is a vector in (mathbb{R}^n). The solution set consists of all vectors (x) that satisfy this equation.

Zero Vector

The zero vector (mathbf{0} in mathbb{R}^n) is always a solution to the equation (Ax 0) because (A mathbf{0} mathbf{0}). This satisfies the requirement that the vector space must contain the zero vector.

Closure under Addition

Suppose (mathbf{x}_1) and (mathbf{x}_2) are two solutions to (Ax 0). Then:

[A mathbf{x}_1 mathbf{x}_2 A mathbf{x}_1 A mathbf{x}_2 mathbf{0} mathbf{0} mathbf{0}]

This shows that (mathbf{x}_1 mathbf{x}_2) is also a solution, demonstrating closure under addition.

Closure under Scalar Multiplication

Let (mathbf{x}) be a solution to (Ax 0), and let (alpha) be a scalar. Then:

[A (alpha mathbf{x}) alpha (A mathbf{x}) alpha mathbf{0} mathbf{0}]

This indicates that (alpha mathbf{x}) is also a solution, showing closure under scalar multiplication.

Conclusion

Since the solution set contains the zero vector, is closed under addition, and is closed under scalar multiplication, it satisfies all the properties of a vector space. Therefore, the solution set of the homogeneous matrix equation (Ax 0) is a vector space.

Vector Space Axioms

A vector space (V) is a set such that for any vectors (mathbf{v}, mathbf{w} in V) and any scalar (alpha), the following three properties hold:

(mathbf{0} in V): The vector space must contain the zero vector. (mathbf{v} mathbf{w} in V): The sum of any two vectors is also in the vector space. (alpha mathbf{v} in V): Any scalar multiple of a vector is also in the vector space.

These axioms are verified for the solution set of the homogeneous matrix equation (Ax 0):

The zero vector is always a solution to (Ax 0). If (mathbf{x}) and (mathbf{y}) are both solutions to (Ax Ay 0), then (mathbf{x} mathbf{y}) is also a solution since (A (mathbf{x} mathbf{y}) A mathbf{x} A mathbf{y} mathbf{0} mathbf{0} mathbf{0}). Any scalar times the zero vector is still the zero vector: (alpha mathbf{0} mathbf{0}).

Therefore, the set of solutions (mathbf{x}) to (Ax 0) forms a vector space.