Basis

A (vector) basis is any Set of $n$ Linearly Independent Vectors capable of generating an $n$-dimensional Subspace of $\Bbb{R}^n$. Given a Hyperplane defined by

$$x_1+x_2+x_3+x_4+x_5 = 0,$$

a basis is found by solving for $x_1$ in terms of $x_2$, $x_3$, $x_4$, and $x_5$. Carrying out this procedure,

$$x_1 = -x_2-x_3-x_4-x_5,$$

so

$$\left[\matrix{x_1 \cr x_2 \cr x_3 \cr x_4 \cr x_5\cr}\right]... ...t] + x_5 \left[\matrix{-1 \cr 0 \cr 0 \cr 0 \cr 1\cr}\right],$$

and the above Vector form an (unnormalized) Basis. Given a Matrix A with an orthonormal basis, the Matrix corresponding to a new basis, expressed in terms of the original ${\bf\hat x}_1, \ldots, {\bf\hat x}_n$ is

$${\hbox{\sf A}}' = \left[\matrix{{\hbox{\sf A}}{\bf\hat x}_1 & \ldots & {\hbox{\sf A}}{\bf\hat x}_n}\right].$$

See also Bilinear Basis, Modular System Basis, Orthonormal Basis, Topological Basis