Euclidean Subspace - Systems of Linear Equations

Systems of Linear Equations

The solution set to any homogeneous system of linear equations with n variables is a subspace of Rn:

$\left\{ \left \in \textbf{R}^n : \begin{alignat}{6} a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; = 0& \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; = 0& \\ \vdots\;\;\; && && \vdots\;\;\; && && \vdots\;\;\; && \vdots\,& \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; = 0& \end{alignat} \right\}.$

For example, the set of all vectors (x, y, z) satisfying the equations

is a one-dimensional subspace of R3. More generally, that is to say that given a set of n, independent functions, the dimension of the subspace in Rk will be the dimension of the null set of A, the composite matrix of the n functions.

Read more about this topic: Euclidean Subspace

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