Detect dependence and use a basis

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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 whether the three vectors are independent. Give a basis and dimension for their span, then give the coordinates of $x=(2,3,5)$ in your basis.

Reveal answer or reference solution

Since $v_3=v_1+v_2$ , the three-vector list is dependent. The vectors $v_1,v_2$ are independent, so $B=(v_1,v_2)$ is a basis and the span has dimension $2$ . Because $x=2v_1+3v_2$ , its $B$ -coordinates are $[x]_B=(2,3)^{\mathsf T}$ . The basis matrix $P_B=[v_1\ v_2]$ performs the change to standard coordinates: $x=P_B[x]_B$ .

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Detect dependence and use a basis

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