Suppose V $\subseteq$ $R^k$ is a subspace of dimension n. What is the dimension of the subspace $V^⊥$that consists of all vectors that are orthogonal to all vectors in V? Given a basis of V, how can we find a basis of $V^⊥$?
question about two sets of orthogonal vectors
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linear-algebra
matrices
vector-spaces
orthogonality
1 Answers
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The dimension of $V^{\perp}$ is $n - k$. One way to see it is to complete a basis $v_1,\dots,v_n$ to a basis $v_1,\dots,v_k$ of $\mathbb{R}^k$ and perform the Gram-Schmidt process on $(v_1,\dots,v_k)$. You'll get vectors $(w_1,\dots,w_k)$ such that $w_1,\dots,w_n$ form an orthonormal basis of $V$ and $w_{n+1},\dots,w_k$ form an orthonormal basis of $V^{\perp}$. This also gives you a way to find an orthonormal basis for $V^{\perp}$.
Alternatively, given a basis $(v_1,\dots,v_n)$ for $V$, you can set up a system of $n$ linear homogeneous equations
$$ \left< x, v_1 \right> = \dots = \left< x, v_n \right> = 0 $$
(where $x = (x_1,\dots,x_k)^T$ is the vector of variables). Solving this system (using row elimination for example) will give you a basis for $V^{\perp}$.