Linear Transformation

What cannot be defined may still be learned and even if we cannot explain the meaning of a concept, as long as it has observable manifestations, we can hope to infer models to predict future observations.  Linear algebra is the branch of mathematics concerning vector spaces (collection of objects called vectors, which may be added together and multiplied by numbers, called scalars) and linear mappings between such spaces.  It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.  Linear equations make a straight line when they are graphed and linear transformations do not change the area of an object.

In plane geometry, a shear mapping or a shear transformation is a linear map that displaces each point in fixed direction, by an amount proportional to its signed distance from a line that is parallel to that direction.  All points along a given line remain fixed while other points are shifted parallel to by a distance proportional to their perpendicular distance that they originated from.  A rotation is a transformation that turns a figure about a fixed point called the center of rotation.   An object and its rotation are the same shape and size, but the figures may be turned in different directions.  Rotations may be clockwise or counterclockwise.  A reflection is a transformation that basically flips a shape over the line of reflection.  Objects are usually reflected across either the x-axis, or the y-axis, or the origin.

There was a trilogy of movies called “The Matrix”, starring Keanu Reeves, but this movie has nothing to do with math.  A matrix in math is like a table or an array, because they are all composed on a grid containing columns and rows, where each location in the grid contains some information.  Matrices differ from tables and spreadsheets, because they include a way of organizing numbers, so data can be visualized for analysis.  A matrix is a rectangular grid and the size of a matrix is defined by the number of rows and columns that it contains.  A matrix with m rows and n columns is called an m × n matrix or m-by-n matrix, while m and n are called its dimensions.  The individual items in a matrix are called its components, elements or entries and they reside in cells.  A matrix with the same number of rows and columns, can be used sometimes to represent a linear transformation from a vector space to itself, such as reflection, rotation, or shearing.

Written for Sammi Cox Author Aspiring December 9, 2017 Weekend Writing Prompt #32 – Reflection.

9 thoughts on “Linear Transformation

  1. I enjoyed reading this. You always have such an interesting take on the prompt, Jim, and I learn something new every time I visit. Thank you 🙂
    Thanks for joining in with the Weekend Writing Prompt. Have a great weekend 🙂

    Liked by 1 person

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