Suppose $U$ is a supspace of $\mathbb{R}^{4}$ defined by $U = \operatorname{span}((1,2,3,-4),(-5,4,3,2))$ Find: orthonormal basis of $U$ and $U^{\perp}$.
I have no issue finding an orthonormal basis of $U$. I would just use the Gram Schmidt Process. $U^{\perp}$ is giving me issues. So what I thought of doing was obtain an orthonormal basis for $U$ and then by the relation of inner product spaces I know that:
Letting $u_{i} \in U$ and $v\in U^{\perp}$ $$<u_1,v> = 0$$ and $$<u_2,v> = 0$$
I would end up with two equations and four unknowns. I debated on if I could pick two of the unknowns and let them be whatever I choose, but that may affect what vectors would be in my subspaces. So I am stuck at this point.
Suggestions ?
Yes by the conditions $\langle u_1,v\rangle = 0$ and $\langle u_2,v\rangle = 0$ we obtain a 2-by-4 system
$$Av=0$$
from which we can find a basis $\{v_1,v_2\}$ for $U^\perp$ which can be orthonormalized by GS process.
Note that solving the system we have $2$ unknowns free since $n=4$ and $\operatorname{rank}(A)=2$ but any choice with $\{v_1,v_2\}$ linearly independent is good to obtain a basis for $U^\perp$.