Vector spaces and spans

Question

Given $V$ vector space on R and $_1,_2 \in V$. We mark $:= \text{span}\{u_1,u_2\}$. Now let $v,w \in U$:

a) Prove that $\text{span}\{v,w\} \subseteq U$;

b) Prove or give counter example $\text{span}\{v,w\} = U$.

Very confused about how to start. any advice? I looked at this for a long time and I'm drawing blank (Just started learning about spans).

Answer 1

How does $\text{span}(v,w)$ look like? Those are exactly the elements of the form $\lambda v + \mu w$, $\lambda,\mu \in F$ where $F$ is a field. Those element are called linear combinations. But since $U$ is a subspace of $V$ we have that any such element is in $U$. For example $ v = \lambda w$ is a counter example in the case that $u_1$ and $u_2$ are linearly independent. Therefore the answer depends on the properties of $u_1$ and $u_2$.

Answer 2

I looked at this for a long time and I'm drawing blank

When one doesn't know how to get started, one can always try to write down the definition for every single concept in the problem.

Exercises:

• What is the definition for $U=\textrm{span}\{u_1,u_2\}$?
• By the definition above, what does $v,w\in U$ mean? Do you see that $v$ and $w$ can be written as linear combinations of $u_1$ and $u_2$?
• To show $\text{span}\{v,w\}\subset U$, note that $A\subset B$ means for every $a\in A$, one has $a\in B$. Now, suppose $z\in\text{span}\{v,w\}$. What does this mean? Can you conclude that $z\in U$?
• Try to finish (a) before asking (b).