Let
$$U = Sp\{(3, 3, 1)\}$$
How can I find the Orthogonal complement ?
I'm not sure how to calculate it.
In the book I'm learning from it's saying that I need to write the vectors of $U$ in $Ax = 0$ where the lines of $A$ are the vectors of $U$.
But since $U$ has only one vector I'm not sure how could this help me to find the orthogonal complement
Best Answer
Since $U$ has only one dimension, it is indeed true that $A$ will have only one line. Hence, the orthogonal complement $U^\perp$ is the set of vectors $\mathbf x = (x_1,x_2,x_3)$ such that \begin{equation} 3x_1 + 3x_2 + x_3 = 0 \end{equation} Setting respectively $x_3 = 0$ and $x_1 = 0$, you can find 2 independent vectors in $U^\perp$, for example $(1,-1,0)$ and $(0,-1,3)$. These generate $U^\perp$ since it is two dimensional (being the orthogonal complement of a one dimensional subspace in three dimensions). Hence, we can conclude that \begin{equation} U^\perp = \operatorname{Span}\{(1,-1,0),(0,-1,3)\}. \end{equation} Note that there would be many (infinitely many) other ways to describe $U^\perp$.