# Orthogonal Bases

## Orthogonal Bases

Definition

Let $V$ be a subspace in $\R^n$.

1. If a basis $B$ for $V$ is an orthogonal set, then $B$ is called an orthogonal basis.
2. If a basis $B$ for $V$ is an orthonormal set, then $B$ is called an orthonormal basis.
Summary

Let $V$ be a subspace in $\R^n$.

1. From any basis $B$ of $V$, the Gram-Schumidt orthogonalization produces an orthogonal basis $B’$ for $V$.

=solution

### Problems

1. Let $S=\{\mathbf{v}_1, \mathbf{v}_2\}$ be the set of the following vectors in $\R^4$.
$\mathbf{v}_1=\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}.$ Find an orthogonal basis of the subspace $\Span(S)$ of $\R^4$.

2. Let $W$ be a subspace of $\R^4$ with a basis
$\left\{\, \begin{bmatrix} 1 \\ 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \\ 1 \end{bmatrix} \,\right\}.$ Find an orthonormal basis of $W$.
(The Ohio State University)

3. Let $A=\begin{bmatrix} 1 & 0 & 1 \\ 0 &1 &0 \end{bmatrix}$.
(a) Find an orthonormal basis of the null space of $A$.
(b) Find the rank of $A$.
(c) Find an orthonormal basis of the row space of $A$.
(The Ohio State University)

4. Let $\mathbf{v}_1=\begin{bmatrix} 2/3 \\ 2/3 \\ 1/3 \end{bmatrix}$ be a vector in $\R^3$. Find an orthonormal basis for $\R^3$ containing the vector $\mathbf{v}_1$.

5. Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ be a set of nonzero vectors in $\R^n$. Suppose that $S$ is an orthogonal set.
(a) Show that $S$ is linearly independent.
(b) If $k=n$, then prove that $S$ is a basis for $\R^n$.