Question 5.5.
Give an orthogonal set of 3 non-zero vectors in \(\mathbb{R}^5\text{.}\)
Section 5.2 Orthogonal Complements
A set of vectors is orthogonal if every pair of distinct vectors in the set is orthogonal.
Let \(W\) be a subspace of an inner product space \(V\text{.}\) The orthogonal complement of \(W\text{,}\) denoted \(W^\bot\text{,}\) is the set of vectors in \(V\) that are orthogonal to every vector in \(W\text{.}\) We read \(W^\bot\) as “\(W\) perp”.
Question 5.6.
Let \(W=span(\{ \colvec{1\\ 1} \})\text{.}\) What is \(W^\bot\text{?}\)
Question 5.7.
Let \(W=span(\{ \colvec{1\\ 0\\ 0},\colvec{0\\ 1\\ 0} \})\text{.}\) What is \(W^\bot\text{?}\)
Question 5.8.
Let \(W=span(\{ f(t) = t \})\) be a subspace of \(C([0,1])\text{.}\) What is \(W^\bot\text{?}\)
Question 5.9.
Prove that if \(W\) is a subspace of an inner product space \(V\text{,}\) then \(W^\bot\) is a subspace of \(V\text{.}\)
