## Elementary Linear Algebra 7th Edition

all the vectors spanned by the vectors $\{(1,0,-\frac{1}{2}),(0,1,\frac{1}{2})\}$ are orthogonal to $u$.
Let $u=(1-1,2)$ and $v=(a,b,c)$ such that $u$ is orthogonal to $v$, that is $$u\cdot v=0 \Longrightarrow a-b+2c=0.$$ Now, assume that, $a=s$ and $b=t$, then we get the solution, $$a=s, \quad b=t, \quad c=-\frac{1}{2}s+\frac{1}{2}t$$ that is, $v=(s,t,-\frac{1}{2}s+\frac{1}{2}t)=s(1,0,-\frac{1}{2})+t(0,1,\frac{1}{2})$ Hence, all the vectors spanned by the vectors $\{(1,0,-\frac{1}{2}),(0,1,\frac{1}{2})\}$ are orthogonal to $u$.