3.6 Orthogonal Complement
Definition 3.9 Given a subspace \(U \subseteq V\), the orthogonal complement \(U^\perp\) contains all vectors in \(V\) orthogonal to every vector in \(U\).
It is easily shown that \(U \cap U^\perp = \{\mathbf{0}\}\), and \(\dim(U) + \dim(U^\perp) = \dim(V)\). Furthermore, any vector \(\mathbf{x} \in V\) can be decomposed as: \[ \mathbf{x} = \sum_{m=1}^{M} \lambda_m \mathbf{b}_m + \sum_{j=1}^{D-M} \psi_j \mathbf{b}_j^\perp. \] In \(\mathbb{R}^3\), a plane and its normal vector illustrate the relationship between a subspace and its orthogonal complement.
Exercises
Exercise 3.39 Show that \(U \cap U^{\perp} = \{ \mathbf{0} \}\).
Exercise 3.40 Given a subspace \(U \subseteq V\), show that \(\dim(U) + \dim(U^\perp) = \dim(V)\).
Exercise 3.41
Find all vectors orthogonal to \(\mathbf{v}_1 = \begin{bmatrix}1\\1\\-1 \end{bmatrix}\) and \(\mathbf{v}_2 = \begin{bmatrix}1\\0\\2 \end{bmatrix}\).Exercise 3.42
What is \(\text{Span}\left\{\begin{bmatrix}1\\1\\-1 \end{bmatrix}, \begin{bmatrix}1\\1\\1\\ \end{bmatrix} \right\}^{\perp}\)?Exercise 3.43
Write \(\mathbf{v} = \begin{bmatrix} 3\\2\\1 \end{bmatrix}\) as a sum of vectors from \(\text{Span} \left\{ \begin{bmatrix}1\\1\\0 \end{bmatrix} \right\}\) and \(\text{Span}\left\{\begin{bmatrix}1\\1\\0 \end{bmatrix} \right\}^{\perp}\).Exercise 3.44
Write \(\mathbf{v} = \begin{bmatrix} 3\\-4\\6 \end{bmatrix}\) as a sum of vectors from \(\text{Span}\left\{\begin{bmatrix}1\\-1\\2 \end{bmatrix},\begin{bmatrix}1\\0\\0 \end{bmatrix} \right\}\) and \(\text{Span}\left\{\begin{bmatrix}1\\-1\\2 \end{bmatrix},\begin{bmatrix}1\\0\\0 \end{bmatrix}\right\}^{\perp}\).