Exercise 595Linear algebra › Vector spaces and subspacesHigh schoolAdded 3 Nov 2026

Span of a set of vectors

#span#linear combinations#generating set

Statement

In $\mathbb{R}^3$ , describe geometrically the span of each set:

(a) $S_1=\{(1,0,0)\}$ ;
(b) $S_2=\{(1,0,0),(0,1,0)\}$ ;
(c) $S_3=\{(1,0,0),(0,1,0),(0,0,1)\}$ ;
(d) $S_4=\{(1,1,0),(2,2,0)\}$ .

Hint

The span is the set of all linear combinations. Look at the dimension.

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(a) $\operatorname{span}(S_1)=\{(\lambda,0,0):\lambda\in\mathbb{R}\}$ : the $x$ -axis.

(b) $\operatorname{span}(S_2)=\{(\lambda,\mu,0):\lambda,\mu\in\mathbb{R}\}$ : the plane $z=0$ .

(c) $\operatorname{span}(S_3)=\mathbb{R}^3$ : the whole space.

(d) The vector $(2,2,0)=2(1,1,0)$ is a multiple of $(1,1,0)$ , so the span is the line $\{(\lambda,\lambda,0):\lambda\in\mathbb{R}\}$ .

Conclusion. The spans are respectively a line, a plane, the whole space, and a line.