Let $V = \mathbb{R}^4$.
Consider the subspace
$$U = \{(a_1,a_2,a_3,a_4) \in \mathbb{R}^4 | a_1 +a_2 +a_3 = 0\} \;of\; V$$
Consider the elements $u_1 =(0,0,0,1)$ and $u_2 =(5,−2,−3,0)$ of $U$. Find another element $u_3 \in U$ such that $\{u1,u2,u3\}$ is a basis of $U$, and prove that it is indeed a basis.
I know the proof of a basis is that the elements must be linearly independent and spans the entire vector space, but how do you do this?
Best Answer
$u_3 = (1,0,-1,0)$ would work. $u_3$ is in $U$, and it is independent from $u_1$ and $u_2$.
And to prove that the vectors span U in a hand-wavy way.
The dimension U is less than 4, since U is a subspace in $R^4$, and clearly there are vectors in $R^4$ that are not in U, and we have 3 independent vectors in U, the dimension of U must be 3 and these vectors span U.