Reading notes

conservatives in America make less sense because America is supposed to be liberal/new

For most Europeans who came to America, the whole purpose of their difficult and dis- ruptive journey to the New World was not to conserve European institutions but to leave them behind and to create something new, often an entirely new life

Three splits of conservatism in America

1. those who are most concerned about economic or fiscal issues, that is, pro-business or “free-enterprise” conservatives
2. those most concerned with religious or social issues, that is, pro-church or “traditional-values” conservatives
3. those most concerned with national-security or defense issues, that is, pro-military or “patriotic” conservatives

Ronald Reagan unified the three conservatism

It was the achievement of Ronald Reagan that he was able in the late 1970s to unite these three different kinds of conservatism into one grand coalition.

Two vectors are considered orthogonal if \(\langle u,v \rangle = 0\), that is, their inner product is \(0\).

See also orthogonality test.

orthogonality and \(0\)

• \(0\) is orthogonal to every vector in \(v\) because \(\langle 0,v \rangle=0\) for every \(v\) because of the properties of inner product
• \(0\) is the only vector orthogonal to itself as, by inner product definiteness, \(\langle v,v \rangle=0\) implies \(v=0\).

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.