Solution Space

The solution space (also called the null space, or kernel) is the vector space which contains the solutions of a linear system. Since linear systems can be can be represented with matrices, a solution space also exists for a coefficient matrix, (by augmenting a zero column matrix to it).

This can be thought of as the more ‘abstract’ version of simply solving a linear system, as it generalises the solution over all vector spaces!

Gauss-Jordan Elimination does not affect the solution space of a matrix.

Definition §

Let $M_{m,n}(\mathbb{F})$ be the set of all matrices with of size $m\times n$ with entries $\in \mathbb{F}$. Then, let a single matrix, $\mathbf{A}$ be an element of the set: $A\in M_{m,n}(\mathbb{F})$. Then the solution space of A is defined by:

S = \{ \boldsymbol{x} \in \mathbb{F}^n| \boldsymbol{Ax} = \boldsymbol{0} \}$$ $S$ is the solution space of $\boldsymbol{A}$ and each vector $\vec{x}$ is a solution of the associated homogenous linear system of $\boldsymbol{A}$. The [Dimension](dimension.md) of $S$ is called the [nullity](Nullity.md) of $\boldsymbol{A}$ i.e. $\text{dim}(S) = \text{nullity}(\boldsymbol{A})$ The nullity is **also the number of non-pivot columns of $\boldsymbol{A}$, when reduced to [RREF](Reduced%20Row%20Echelon%20Form.md). ### Obtaining the [basis](Basis.md) of the Solution Space