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Matrices
Matrices and matrix multiplication reveal their essential features when related to //linear transformations//, also known as //linear maps//. A real //m//-by-//n// matrix **A** gives rise to a linear transformation **R**//n// → **R**//m// mapping each vector **x** in **R**//n// to the (matrix) product **Ax**, which is a vector in **R**//m//. Conversely, each linear transformation //f//: **R**//n// → **R**//m// arises from a unique //m//-by-//n// matrix **A**: explicitly, the (//i//, //j//)-entry of **A** is the //i//th coordinate of //f//(**e**//j//), where **e**//j// = (0,...,0,1,0,...,0) is the unit vector with 1 in the //j//th position and 0 elsewhere. The matrix **A** is said to represent the linear map //f//, and **A** is called the //transformation matrix// of //f//. For example, the 2×2 matrix

﻿Multiplying Matrices When you multiply matrices you always do something known as dipping and diving.

