# Vectors - Spanning sets

## Homework Statement

Consider a set of vectors:

S = {v$_{1}$, v$_{2}$, v$_{3}$, v$_{4}$$\subset$ ℝ$^{3}$

a) Can S be a spanning set for ℝ$^{3}$? Give reasons for your answer.
b) Will all such sets S be spanning sets? Give a reason for your answer.

## The Attempt at a Solution

a) Yes, because a linear combination of these vectors can form any given vector in ℝ$^{3}$.

b) Yes, don't really know a reason besides something similar to the one above, I can't see why not.

I'm not really sure of these answers, can anyone confirm this, or given any insight to understand this better?

Cheers

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HallsofIvy
Homework Helper
What about, {(1, 1, 0), (1, 2, 0), (3, 1, 0)}?

Well I guess since the 3rd elements are zero, you can't form a vector in ℝ$^{3}$ which has a non-zero 3rd element.

So part b is definitely a no. However part a "can" be since they haven't explicitly defined the vectors, so it's possible given that zero/non-zero condition, is that the only reason?

HallsofIvy