Here is the question I am on.
Let $V=\mathbb{R^5}$ be a vector space over $\mathbb{R}$ and let
$U=\{(a_1,a_2,a_3,a_4,a_5)\in \mathbb{R^5}|a_1+2a_2+a_3+a_5=0\}$.Let $u_1=(1,-1,1,0,0)$ and $u_2=(0,0,-1,0,1)$. Find vectors
$v_1,…,v_k\in U$ such that $\{u_1,u_2,v_1,…,v_k\}$ is a basis for
$U$.
Here is what I have done:
We know we can extend the set to form a basis by simply adding a vector that is not in the span of $u_1,u_2$ such a vector would be $v_1=(0,0,0,1,0) \in U$ now we know $\{u_1,u_2,v_1\}$ are a linearly independent set of vectors. My question is now how to proceed. My instinct is telling me that $U$ is a $4$ dimensional subspace but I'm still not sure on how to proceed. I either need to show $\{u_1,u_2,v_1\}$ spans $U$ or find another vector not in the span of $\{u_1,u_2,v_1\}$ both of which I'm finding tricky.
Any help?
Best Answer
You can try finding other solutions to the original condition and seeing if they a linearly independent.
For example, $a_1=2, \, a_2=-1 $ may be a solution.