#linear transformation

[peeterjoot]

Bivector transformation, and reciprocal frame for column vectors of a transformation

[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…

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[mikeo56]

Linear Transformations

Linear Algebra is one of the most important basic areas in Mathematics, having at least as great an impact as Calculus, and indeed it provides a signi¯cant part of the machinery required to generalize Calculus to vector-valued functions of many variables. Unlike many algebraic systems studied in Mathematics or applied within or outwith it, many of the problems studied in Linear Algebra are amenable to systematic and even algorithmic solutions, and this makes them implementable on computers { this explains why so much calculational use of computers involves this kind of algebra and why it is so widely used. Many geometric topics are studied making use of concepts from Linear Algebra, and the idea of a linear transformation is an algebraic version of geometric transformation. Finally, much of modern abstract algebra builds on Linear Algebra and often provides concrete examples of general ideas.

[mbastuffposts]

Linear transformation and its effects

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…

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[blog-krish-blog]

Principal Component Analysis (PCA) Explained with Examples

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…

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[studyingwithjessie]

Functions and Linear Transformations

[matan-matika]

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.

Another cool property is that if λ is less than 1, then T repels vectors from v, and if λ is greater than 1, then it attracts T.

[matan-matika]

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).

Those two lines? That’s the topic of this post.

[matan-matika]

I am in the process of writing a post about eigenvectors. I taught myself how to use Mathematica over the weekend, and the actual GIF came out cool, but I am still trying to find the writing style that explains what is happening well enough while still being fairly accessible.