# Tagged: orthogonal basis

## Problem 716

Using Gram-Schmidt orthogonalization, find an orthogonal basis for the span of the vectors $\mathbf{w}_{1},\mathbf{w}_{2}\in\R^{3}$ if
$\mathbf{w}_{1} = \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} ,\quad \mathbf{w}_{2} = \begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix} .$

## Problem 715

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 $V$.

## Problem 602

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, Linear Algebra Midterm)

## Problem 600

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 $\mathbf{v}_1$.

## Problem 539

Consider the $2\times 2$ real matrix
$A=\begin{bmatrix} 1 & 1\\ 1& 3 \end{bmatrix}.$

(a) Prove that the matrix $A$ is positive definite.

(b) Since $A$ is positive definite by part (a), the formula
$\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans} A \mathbf{y}$ for $\mathbf{x}, \mathbf{y} \in \R^2$ defines an inner product on $\R^n$.
Consider $\R^2$ as an inner product space with this inner product.

Prove that the unit vectors
$\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \end{bmatrix} \text{ and } \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ are not orthogonal in the inner product space $\R^2$.

(c) Find an orthogonal basis $\{\mathbf{v}_1, \mathbf{v}_2\}$ of $\R^2$ from the basis $\{\mathbf{e}_1, \mathbf{e}_2\}$ using the Gram-Schmidt orthogonalization process.

## Problem 356

(a) 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 $\R^4$.

(b) Let $T:\R^2 \to \R^3$ be a linear transformation such that
$T(\mathbf{e}_1)=\mathbf{u}_1 \text{ and } T(\mathbf{e}_2)=\mathbf{u}_2,$ where $\{\mathbf{e}_1, \mathbf{e}_2\}$ is the standard unit vectors of $\R^2$ and
$\mathbf{u}_1=\begin{bmatrix} 5 \\ 1 \\ 2 \end{bmatrix} \text{ and } \mathbf{u}_2=\begin{bmatrix} 8 \\ 2 \\ 6 \end{bmatrix}.$ Then find
$T\left(\, \begin{bmatrix} 3 \\ -2 \end{bmatrix} \,\right).$