Sort each set of vectors in ℝ³: linearly independent or dependent?

Physics I / Vector Spaces and Subspaces

Practice question · Sort into groups

Sort each set of vectors in $\mathbb{R}^{3}$ : linearly independent or dependent?

Groups: Independent · Dependent

Any four vectors in $\mathbb{R}^{3}$
$(1,1,0),\ (0,1,1)$
$(1,0,0),\ (0,1,0),\ (0,0,1)$
$(1,2,3),\ (2,4,6)$

Hints

Dependent means at least one vector is redundant, expressible from the others.
$(2,4,6)$ is exactly twice $(1,2,3)$ . And $\mathbb{R}^{3}$ has only 3 dimensions of room, a fourth vector must fall inside the span of the first three.

Show the answer

Independent: $(1,0,0),\ (0,1,0),\ (0,0,1)$ , $(1,1,0),\ (0,1,1)$

Dependent: $(1,2,3),\ (2,4,6)$ , Any four vectors in $\mathbb{R}^{3}$

Why

Independence = no redundancy = each vector opens a genuinely new direction. Three dimensions can host at most three independent directions, so four vectors are automatically dependent, a counting law with deep consequences (rank, basis, dimension).

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Sort each set of vectors in $\mathbb{R}^{3}$ : linearly independent or dependent?