[Click here for a PDF version of this and previous two posts] The author of a book draft I am reading pointed out if a vector transforms as \begin{equation}\label{eqn:adjoint:760} \Bv \rightarrow M \Bv, \end{equation} then cross products must transform as \begin{equation}\label{eqn:adjoint:780} \Ba \cross \Bb \rightarrow \lr{ \textrm{adj}\, M }^\T \lr{ \Ba \cross \Bb }. \end{equation} Bivectors…

Linear Transformations

Linear transformation and its effects

Learning Objectives

Define a linear transformation

Compute the mean of a transformed variable

Compute the variance of a transformed variable

This section covers the effects of linear transformations on measures of central tendency and variability. Let’s start with an example we saw before in the section that defined linear transformation: temperatures of cities. Table 1 shows the temperatures of…

Principal Component Analysis (PCA) Explained with Examples

Any machine learning model building task begins with a collection of data vectors wherein each vector consists of a fixed number of components. These components represent the measurements, known as attributes or features, deemed useful for the given machine learning task at hand. The number of components, i.e. the size of the vector, is termed as the dimensionality of the feature space. When the…

Eigenvectors

A linear transformation T that stays in the same vector space could have some points that stay relatively fixed. A fixed vector v would have T(v)=v. Many transformations only have the 0 vector as their only fixed point.

But, what happens in a lot is that there are certain vectors that get scaled by the transformation. That is, there exists a scalar λ such that T(v)= λv. Since all scalars are unaffected by T, this same λ works for all scalings of v. If v has this property, it is an eigenvector for T, and λ is an eigenvalue.

A transformation can have multiple independent eigenvectors.

Linear Transformations

Linear transformations are functions that bring vectors in one vector space into vectors in another. There is a lot we could say about them, but the two important features are that if T is a transformation, u and v are vectors, and a is a scalar, then T must preserve two features: T(u + v)=T(u) + T(v) and T(av) = aT(v).

You can do all sorts of interesting transformations that keep these features: switching vectors around, stretching, rotating, and more.

We call them linear because these two features mean that lines get preserved. Why? Because u and u+av for all values of a make a line, and if we apply the transformation, T(u+av)=T(u)+aT(v), which makes all of these on the same line as T(u).