Must vectors span the space?

Question

So i have this question :

Four vectors in $\mathbb{R}^3$ must span all the space. Is this true or false?

I think it is true. Vectors must always span all of the space. Am i correct?

Answer 1

It is false. Counter examples are e.g. $$ \{(1,0,0),(-1,0,0),(2,0,0),(-2,0,0)\} $$ which are collinear and so only span a $1$ dimensional subspace of $\mathbb{R}^3$.

Or $$ \{(1,0,0),(-1,0,0),(0,1,0),(0,-1,0)\} $$ which spans a $2$ dimensional subspace of $\mathbb{R}^3$.

To span $\mathbb{R}^3$ you need at least 3 linearly independent vectors such as $$ \{(1,0,0),(0,1,0),(0,0,1)\} $$ Adding a fourth vector is fine for spanning the space, but it is redundant.

Answer 2

If you have a set of vectors with a subset that is linearly indepent, then the set spans the space. if the number of linearly independt vectors in your set is less than the dimension of the space, then the set does not span the space.

Answer 3

What's true is that a set of vectors spans $\mathbb{R}^3$, then the set consists of at least three elements. There even exist infinite subsets of $\mathbb{R}^3$ that don't span it.

Let $U$ be a proper subspace of $\mathbb{R}^3$, $U\ne\{0\}$. Then you can pick as many elements as you want from $U$ (which is an infinite set) and they will not span $\mathbb{R}^3$, because the span is a subspace of $U$.