Dimension of an affine subspace

L. El Ghaoui

Dimension of an affine subspace

The set ${\bf L}$ in $\mathbb{R}^3$ defined by the linear equations

$x_1 -13 x_2 + 4x_3 =2, \hspace{0.1in} 3x_2 - x_3 = 9$

is an affine subspace of dimension $1$ . The corresponding linear subspace is defined by the linear equations obtained from the above by setting the constant terms to zero:

$x_1 -13 x_2 + 4x_3 =0, \hspace{0.1in} 3x_2 - x_3 = 0$

We can solve for $x_3$ and get $x_1 = x_2, x_3 = 3x_2$ . We obtain a representation of the linear subspace as the set of vectors $x \in \mathbb{R}^3$ that have the form

(1) $\begin{align*} x := \begin{pmatrix} 1 \\ 1 \\ 3 \end{pmatrix} \end{align*}t,$

for some scalar $t=x_2$ . Hence the linear subspace is the span of the vector $u:=(1, 1, 3)$ , and is of dimension $1$ .

We obtain a representation of the original affine set by finding a particular solution $x^0$ , by setting say $x_2 = 0$ and solving for $x_1, x_3$ . We obtain

(2) $\begin{align*} x := \begin{pmatrix} 38 \\ 0 \\ -9 \end{pmatrix} \end{align*}.$

The affine subspace ${\bf L}$ is thus the line $x^0 + {\bf span}(u)$ , where $x^0, u$ are defined above.

Dimension of an affine subspace

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