TITLE: Is this a linear algebra problem?
QUESTION [0 upvotes]: I'm solving this question, but I don't know what it is.

Consider a plane $x - y + z = 0$, and a point $b = (1, 2, 0)^T$
a) Find a basis of this plane, and a basis of orthogonal subspace to this plane.
b) Find the closest point $\hat{b}$ on the plane to b.
c) What is the error between b and $\hat{b}$?

Is this a linear algebra problem? What's a basis of orthogonal subspace and closest point $\hat{b}$? What's the error between b and $\hat{b}$??

REPLY [1 votes]: (a) To find a basis for the plane $x-y+z = 0$, you could solve this equation in terms of $x$ and $z$: $y= x+z$. Then the set of vectors of your plane could be described as:
$$
V = \left\{ (x, x+z, z) \ \vert \ x,z \in \mathbb{R}   \right\} \ .
$$
From this description it's easy to find a basis for your plane $V$: it will have two vectors, say $u,v \in \mathbb{R}^3$:
$$
V = [u,v] \ .
$$
Then, the orthonormal subspace to your plane is a straight line, generated by the cross product of $u$ and $v$
$$
V^\bot = [u \times v] \ .
$$
(b) The closest point $\hat{b}$ to $b$ in the plane $V$ is the orthogonal projection of $b$ onto $V$. In this case, you don't really need the formulae you'll see in Wikipedia, because it's just the intersection of $V$ with the straight line with direction $u\times v$ that passes through $b$. That is, you have to solve the system of linear equations
$$
\begin{align}
x - y +z &= 0    \\
(x,y,z) &= (1,2,0) + \lambda u\times v  \ .
\end{align}
$$
(c) The "error" between $b$ and $\hat{b}$ is the same as the distance from $b$ to $\hat{b}$; that is, $\| b - \hat{b} \|$.
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