The nonviolence movement a method of protest which is developed by Mahatma Ghandi and leveraged by Martin Luther King in the civil rights movement.

The idea is to achieve civil disobedience and allowing oneself to be punished so egregiously without inciting violence so at to elicit sympathy across the nation.

The civil rights movement leveraged training tactics and training to ensure its participants would be completely nonviolent and so elicit the correct response.

The norm is the “length” of a vector, defined generally using the inner product as:

\begin{equation} \|v\| = \sqrt{\langle v,v \rangle} \end{equation}

additional information

properties of the norm

1. \(\|v\| = 0\) IFF \(v=0\)
2. \(\|\lambda v\| = |\lambda|\|v\|\)

Proof:

1. By definition of an inner product, \(\langle v,v \rangle = 0\) only when \(v=0\)
2. See algebra:

\begin{align} \|\lambda v\|^{2} &= \langle \lambda v, \lambda v \rangle \\ &= \lambda \langle v, \lambda v \rangle \\ &= \lambda \bar{\lambda} \langle v,v \rangle \\ &= |\lambda |^{2} \|v\|^{2} \end{align}

See Gaussian distribution

for small equations of Linear Regression, we can solve it using normal equation method.

Consider \(d\) dimensional feature and \(n\) samples of data. Remember, including the dummy feature, we have a matrix: \(X \in \mathbb{R}^{n \times \qty(d+1)}\) and a target \(Y \in \mathbb{R}^{n}\).

Notice:

\begin{equation} J\qty(\theta) = \frac{1}{2} \sum_{i=1}^{n} \qty(h_{\theta} \qty(x^{(i)}) - y^{(i)})^{2} \end{equation}

and \(h = X \theta\), we we can write:

\begin{equation} J(\theta) = \frac{1}{2} \qty(X \theta - y)^{T} \qty(X \theta - y) \end{equation}