I am doing a question on Linear combinations to revise for a linear algebra test. I do not have access to the solutions therefore I am not sure if I am corrects or if my intuitions are correct, also I am stuck in a few places. Would be great if someone can help me out.
$u = (1,2,1)$ and $v=(2,-1,0)$
1) The vector $w$ is a linear combination of the vectors ${u, v}$ if:
$w = au + bv,$ for some $a,b \in \mathbb{R} $ (is this correct?)
2) The span of two vectors $u, v ∈ \mathbb{R}^3$ is the set of vectors:
span{u,v} = {a(1,2,1) + b(2,-1,0)} (is this correct?)
3) Write down a geometric description of the span of two vectors $u, v ∈ \mathbb{R}^3$.
I dont understand what is required here. Please help.
4) Is it possible to find two vectors whose span is a plane that does not pass through the origin? If not, explain why not. If so, find two vectors that achieve this.
Same as three. I have no idea.
Any help would be greatly appreciated
Best Answer
1) Is correct, see the definition of linear combination
2) Yes, maybe you'll see the notation $\langle\{u,v\}\rangle$ for the span of $u$ and $v$ and it's definition
$$ \langle\{u,v\}\rangle = \left\{w\in \mathbb{R}^3\; : \; w = a u+bv, \; \; a,b\in\mathbb{R} \right\}$$
3) The span of two vectors in $\mathbb{R}^3$
4) No, the span of $u,v$ is a vector subspace of $\mathbb{R}^3$ and every vector space contains the zero vector, in this case $(0,0,0)$