- #1

Unredeemed

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## Homework Statement

Let V be the finite-dimensional vector space of 2 x 2 matrices with real entries.

State, without proof, the dimension of V as a real vector space. (done)

A real-valued function V x V is defined by:

<A|B> = tr((A^t)B) where A^t is the transpose of the matrix A.

Show that <A|B> defines an inner product on V. (done)

Show that, with respect to this inner product, any symmetric matrix is orthogonal to any antisymmetric matrix, and that the identity matrix is orthogonal to any matrix whose trace is zero. (done)

Find an orthonormal basis of V. (bit I can't do)

## Homework Equations

## The Attempt at a Solution

I thought I could just write down:

\begin{equation}

\begin{pmatrix}

1 & 0 \\

0 & 1

\end{pmatrix}

\end{equation}

\begin{equation}

\begin{pmatrix}

1 & 0 \\

0 & -1

\end{pmatrix}

\end{equation}

\begin{equation}

\begin{pmatrix}

0 & -1 \\

1 & 0

\end{pmatrix}

\end{equation}

\begin{equation}

\begin{pmatrix}

0 & 1 \\

1 & 0

\end{pmatrix}

\end{equation}

And then show that it's an orthonormal basis. But, this obviously doesn't use what I've already proved. How can I deduce it from what I already have?

Cheers.