Basis in high dimension

L. El Ghaoui

Basis in high dimension

The set of three vectors in $\mathbb{R}^3$ :

(1) $\begin{align*} x_1:= \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \end{align*},$

(2) $\quad \begin{align*} x_2:= \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix} \end{align*},$

(3) $\quad \begin{align*} x_3 := \begin{pmatrix} 3 \\ 3 \\ 3 \end{pmatrix} \end{align*}$

is not independent, since $x_1-x_2+x_3=0$ , and its span has dimension $2$ . Since $x_1, x_2$ are independent (the equation $\lambda_1 x_1 + \lambda_2 x_2 = 0$ has $\lambda = 0$ as the unique solution), a basis for that span is, for example, $\{ x_1, x_2 \}$ . In contrast, the collection $\{ x_1, x_2, x_3 - e_1 \}$ spans the whole space $\mathbb{R}^3$ , and thus forms a basis of that space.

Basis in high dimension

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