Basis, dimension, and coordinates

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Problem

In $\mathbb{R}^3$ , let $v_1 = (1,0,1)$ , $v_2 = (0,1,1)$ , and $v_3 = (1,1,2)$ . Determine a basis and dimension for their span. Express $x = (2,3,5)$ in your chosen basis.

Reveal answer or reference solution

Since $v_3 = v_1 + v_2$ , the set $\{v_1,v_2\}$ is a basis and the span has dimension $2$ . Since $x = 2v_1 + 3v_2$ , its coordinates in that basis are $(2,3)$ .

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Appears in

Linear Algebra Practice