A linear transformation between two vector spaces and is a map W’ border=’0′ height=’14’ width=’60’> such that the following hold: A linear transformation may or may not be injective or surjective. When and have the same dimension, it is possible for to be invertible, meaning there exists a such that . It is always the case that . Also, a linear transformation always maps lines to lines (or to zero). The main example of a linear transformation is given by matrix multiplication. Given an matrix , define , where is written as a column vector (with coordinates). For example, consider When and are finite dimensional, a general linear transformation can be written as a matrix multiplication only after specifying a vector space basis for and . When and have an inner product, and their vector space bases, and , are orthonormal, it is easy to write the corresponding matrix . In particular, . Note that when using the standard basis for and , the th column corresponds to the image of the th standard basis vector. When and are infinite dimensional, then it is possible for a linear transformation to not be continuous. For example, let be the space of polynomials in one variable, and be the derivative. Then , which is not continuous because 0′ border=’0′ height=’14’ width=’53’> while does not converge. Linear two-dimensional transformations have a simple classification. Consider the two-dimensional linear transformation This gives two fixed points, which may be distinct or coincident. The fixed points are classified as follows. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. Source.