I can figure out the first part of the question just fine, by using the Gram-Schmidt process, except I am having trouble with finding an orthonormal basis for the orthogonal complement of U, or "U perp".
Would I need to find the orthonormal basis for the space U and then use that basis to find the orthonormal basis for its orthogonal complement? Or is it possible to find the orthonormal basis for U-perp just by finding the orthogonal complement of the set of vectors in the picture then using Gram-Schmidt process on those?
Again, I can find the orthonormal basis for the space U just fine, except I have no idea where to go from here.
EDIT: Here is the orthonormal basis for the space U generated by the three vectors pictured.
EDIT2: I solved for the null-space of the three vectors and came up with a fourth vector, then I applied Gram-Schmidt to the fourth vector with respect to the first three and obtained this as a result, would this one vector here be the orthonormal basis of the orthogonal complement of U? V4 is the vector obtained from solving for the null-space of the first three.
Extend the given basis for $U$ to a basis for $\mathbb{R}^4$ before applying Gram-Schmidt to the entire thing. Then the first three vectors of the result give you a basis for $U$ and the last, being orthogonal to all three, gives you a basis for $U^\perp$.