Orthogonal Bases
Definition
Let $V$ be a subspace in $\R^n$.
- If a basis $B$ for $V$ is an orthogonal set, then $B$ is called an orthogonal basis.
- If a basis $B$ for $V$ is an orthonormal set, then $B$ is called an orthonormal basis.
Summary
Let $V$ be a subspace in $\R^n$.
- From any basis $B$ of $V$, the Gram-Schumidt orthogonalization produces an orthogonal basis $B’$ for $V$.
=solution
Problems
- Using Gram-Schmidt orthogonalization, find an orthogonal basis for the span of the vectors $\mathbf{w}_{1},\mathbf{w}_{2}\in\R^{3}$ if
\[
\mathbf{w}_{1}
=
\begin{bmatrix}
1 \\ 0 \\ 3
\end{bmatrix}
,\quad
\mathbf{w}_{2}
=
\begin{bmatrix}
2 \\ -1 \\ 0
\end{bmatrix}
.
\] - Let
\[
\mathbf{v}_{1}
=
\begin{bmatrix}
1 \\ 1
\end{bmatrix}
,\;
\mathbf{v}_{2}
=
\begin{bmatrix}
1 \\ -1
\end{bmatrix}
.
\] Let $V=\Span(\mathbf{v}_{1},\mathbf{v}_{2})$. Do $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ form an orthonormal basis for $V$? If not, then find an orthonormal basis for $V$. -
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 $\R^4$. - 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) -
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) - 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 $\mathbf{v}_1$. - 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 $\R^n$.