Matrix (plural: matrices) is a rectangular array or a Vector of numbers, symbols, or expressions, arranged in rows and columns.

For example, the dimensions of the matrix below are m × n (read "like two by three"), because there are m rows and n columns:


The m rows are horizontal and the n columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts. For example, a2,1 represents the element at the second row and first column of a matrix A.

Matrices can be thought of as transforming space, and understanding how this work is crucial for understanding many other ideas that follow in linear algebra. Choosing just one topic that makes all of the others in linear algebra start to click and which too often goes unlearned the first time a student takes linear algebra it would be this one the idea of a linear transformation and its relation to matrices. A transformation is a Function. So a linear Function is a Function that takes a vector input and outputs a vector.

A Linear transformation implies there must be no curves (only lines) and the origin must remain fixed. Think of linear transformations as keeping gridlines parallel and evenly spaced.

Basis vectors define the coordinate system or origin. Commonly iHat (x-axis) and jHat (y-axis) are used.

A linear transformation requires that you perform matrix multiplication of the vector by the changes to the Basis vectors (iHat, jHat) Any linear transformation can be described as iHat and jHat this is because any other vector could be described as a linear combination of those basis vectors a vector with coordinates x, y is x times iHat plus y times jHat

"Specialty" Matrix#

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