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 $\R^n$.