## The Inner Product on $\R^2$ induced by a Positive Definite Matrix and Gram-Schmidt Orthogonalization

## 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.