A list of vectors is orthonormal if each vector is orthogonal to every other vector, and they all have norm 1.

In other words:

\begin{equation} \langle e_{j}, e_{k} \rangle = \begin{cases} 1, j = k\\ 0, j \neq k \end{cases} \end{equation}

The vectors should inner-product with itself to \(1\), and be orthogonal to all others.

Additional Information

orthonormal basis

See also orthonormal basis

Norm of an Orthogonal Linear Combination

\begin{equation} \| a_1e_1 + \dots + a_{m}e_{m} \|^{2} = |a_1|^{2} + \dots + |a_{m}|^{2} \end{equation}

An Orthonormal basis is defined as a basis of a finite-dimensional vector space that’s orthonormal.

Additional Information

orthonormal list of the right length is a basis

An orthonormal list is linearly independent, and linearly independent list of length dim V are a basis of V. \(\blacksquare\)

Writing a vector as a linear combination of orthonormal basis

According to Axler, this result is why there’s so much hoopla about orthonormal basis.

Result and Motivation

For any basis of \(V\), and a vector \(v \in V\), we by basis spanning have:

The OTC Markets/pink sheets are an unregulated group of Financial Markets, where many of the Penny stocks are.

action outcomes are uncertain

Polynomial Time \(P\) vs Non-Polynomial Time \(NP\)

• \(P\) are the problems that can be efficiently solved
• \(NP\) are the problems where proposed solutions can be efficiently verified

so! is \(P=NP\)?

If this is true, there are some consequences:

• proving can be automated
• cryptography would be able to be automated easily

anywhere there is a problem can be globally optimized. This is too good to be true, so probably \(P \neq NP\).