Determine which set span $\mathbb{R^3}$

Question

Let $v_1,v_2,v_3$ be vectors in $\mathbb{R^3}$ such that $\langle v_1,v_2,v_3\rangle=\mathbb{R^3}$

Determine which of the following sets span $\mathbb{R^3}$

i)$S=\{v_1,v_2\}$

ii)$T=\{v_1-v_2,v_2-v_3,v_3-v_1\}$

Here how to relate the sets given with the spannning set given?

Answer 1

Hints:

(i) A spanning set of an $n$-dimensional vector space must at least contain $n$ linearly independent vectors.

(ii) These vectors are not linearly independent (what happens if you add them all up?).

Answer 2

One can check the dimension of space in these cases. For example in the first case the dimension can be utmost $2$ where as $\mathbb{R^3}$ clearly has dimension $3$. To find the dimension one has to find all the linearly independent vectors. $T$ also does not span $\mathbb{R^3}$ since all are not linearly independent. So now consider any linear combination of $v_1-v_2$ and $v_2-v_3$ is of form $$av_1+(b-a)v_2-bv_3$$ If this is zero implies $a=0$ and $b=0$

Here we know all the subspaces of $\mathbb{R^3}$ ie they are either a point, a line, a plane or the whole $\mathbb{R^3}$. So if there are two linearly independent vectors they will form a plane. If there is one non zero vector it will form a line. If there are three linearly independent vectors then they will form the whole space.

I thank Casteels for pointing out the error in my initial answer