25.1: Orthogonal and Orthonormal

Last updated
Save as PDF

Page ID: 69406

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Definition

A set of vectors is said to be orthogonal if every pair of vectors in the set is orthogonal (the dot product is 0). The set is orthonormal if it is orthogonal and each vector is a unit vector (norm equals 1).

Result: An orthogonal set of nonzero vectors is linearly independent.

Definition

A basis that is an orthogonal set is called an orthogonal basis. A basis that is an orthonormal set is called an orthonormal basis.

Result: Let \(\{u_1,\dots,u_n\}\) be an orthonormal basis for a vector space \(V\). Then for any vector \(v\) in \(V\), we have \(v=(v\cdot u_1)u_1+(v\cdot u_2)u_2 +\dots + (v\cdot u_n)u_n\).

Definition

A square matrix is orthogonal if \(A^{-1} = A^{\top}\).

Result: Let \(A\) be a square matrix. The following three statements are equivalent.

\(A\) is orthogonal.
The column vectors of \(A\) form an orthonormal set.
The row vectors of \(A\) form an orthonormal set.
\(A^{-1}\) is orthogonal.
\(A^{\top}\) is orthogonal.

Result: If \(A\) is an orthogonal matrix, then we have \(\left| A \right| = \pm 1\).

Consider the following vectors \(u_1\), \(u_2\), and \(u_3\) that form a basis for \(R^3\).

\[ u_1 = (1,0,0) \nonumber \]

\[ u_2 = (0, \frac{1}{\sqrt(2)}, \frac{1}{\sqrt(2)}) \nonumber \]

\[ u_3 = (0, \frac{1}{\sqrt(2)}, -\frac{1}{\sqrt(2)}) \nonumber \]

Do This

Show that the vectors \(u_1\), \(u_2\), and \(u_3\) are linearly independent

(HINT: see the pre-class for 12 Pre-Class Assignment - Matrix Spaces).

Question 1

How do you show that \(u_1\), \(u_2\), and \(u_3\) are orthogonal?

Question 2

How do you show that \(u_1\), \(u_2\), and \(u_3\) are normal vectors?

Do This

Express the vector \(v=(7,5,−1)\) as a linear combination of the \(u_1\), \(u_2\), and \(u_3\) basis vectors:

# Put your answer here