Kernel (matrix) - Definition

Definition

The kernel of an m × n matrix A with coefficients in a field K (typically the field of the real numbers or of the complex numbers) is the set

where 0 denotes the zero vector with m components. The matrix equation Ax = 0 is equivalent to a homogeneous system of linear equations:

$\mathbf{A}\textbf{x}=\textbf{0} \;\;\Leftrightarrow\;\; \begin{alignat}{6} a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; = 0& \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; = 0& \\ \vdots\;\;\; && && \vdots\;\;\; && && \vdots\;\;\; && \vdots\,& \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; = 0. & \end{alignat}$

From this viewpoint, the null space of A is the same as the solution set to the homogeneous system.

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