The solutions to a homogeneous system of linear equations is a vector space.
Created almost 9 years ago, updated 5 months ago
ID24101
Theorem: The set of solutions to a homogeneous system of linear equations (A\mathbf{x} = \mathbf{0}) is a vector space (a subspace of (\mathbb{F}^n), where (n) is the number of variables).
Proof: Let (S = {\mathbf{x} : A\mathbf{x} = \mathbf{0}}).
If (\mathbf{x}, \mathbf{y} \in S), then (A(\mathbf{x} + \mathbf{y}) = A\mathbf{x} + A\mathbf{y} = \mathbf{0} + \mathbf{0} = \mathbf{0}), so (\mathbf{x} + \mathbf{y} \in S).
If (\mathbf{x} \in S) and (c \in \mathbb{F}), then (A(c\mathbf{x}) = cA\mathbf{x} = c\mathbf{0} = \mathbf{0}), so (c\mathbf{x} \in S).
This space is the null space of (A), denoted ( ext{null}(A)) or (\ker(A)).
Example: The solutions to (x + y + z = 0) form a plane through the origin in (\mathbb{R}^3), which is a 2-dimensional subspace.
**Theorem:** The set of solutions to a homogeneous system of linear equations \(A\mathbf{x} = \mathbf{0}\) is a vector space (a subspace of \(\mathbb{F}^n\), where \(n\) is the number of variables).
**Proof:** Let \(S = \{\mathbf{x} : A\mathbf{x} = \mathbf{0}\}\).
- If \(\mathbf{x}, \mathbf{y} \in S\), then \(A(\mathbf{x} + \mathbf{y}) = A\mathbf{x} + A\mathbf{y} = \mathbf{0} + \mathbf{0} = \mathbf{0}\), so \(\mathbf{x} + \mathbf{y} \in S\).
- If \(\mathbf{x} \in S\) and \(c \in \mathbb{F}\), then \(A(c\mathbf{x}) = cA\mathbf{x} = c\mathbf{0} = \mathbf{0}\), so \(c\mathbf{x} \in S\).
This space is the *null space* of \(A\), denoted \( ext{null}(A)\) or \(\ker(A)\).
**Example:** The solutions to \(x + y + z = 0\) form a plane through the origin in \(\mathbb{R}^3\), which is a 2-dimensional subspace.