Orthogonal complement of a line in $\mathbb{R}^2$ spanned by $(1,2)$
Have a subspace $W$ of $\mathbb{R}^2$, where $W$ is spanned by $\{(1,2)\}$. Determine $W^\perp$.
Well then, clearly $W$ is a line in a 2D space. So I guess $W^\perp$ is a line, too.
$W^\perp$ are all the vectors in $\mathbb{R}^2$ that are orthogonal to any vector of $W$.
Since $W$ is a line, any vector in $W$ is a scalar product of $(1,2)$. So, $W^\perp$ is composed by vectors that are orthogonal to $(1,2)$:
$$W^\perp = \{(a,b) \ \ : \ \ (a,b)\cdot (1,2) = 0\}$$
So we have a system of equations:
$$\begin{cases} a \cdot 1 = 0\\ b \cdot 2 = 0 \end{cases}$$
Therefore
$$(a,b) = (0,0)$$
So
$$W^\perp=span\{(0,0)\}$$
... This isn't right... Shouldn't it be $(-2,1)$ or $(2,-1)$?
$\endgroup$1 Answer
$\begingroup$\begin{cases} a \cdot 1 = 0\\ b \cdot 2 = 0 \end{cases} should be $$a\times 1+b\times 2 =0$$ Therefore, you've got $(-2\alpha ,\alpha)$ where $\alpha\in \mathbb{R}$
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