Matrices

If a vector is a “point” and a linear transformation is an “action”, then a matrix is the table that connects the two. Multiplying a matrix by a vector gives a new vector, and multiplying a matrix by a matrix gives a new transformation — almost every operation in linear algebra ultimately reduces to these two.

This chapter starts from the basic concepts of matrices, but it does not stop at “how do I compute it”; it always answers “what does the computed result mean geometrically”. You will see that each column of a matrix is really the position of a basis vector after the transformation, and that single sentence is the key to understanding matrices. It then introduces the inverse matrix as the reverse operation of a transformation, and the adjugate matrix as a shortcut for computing the inverse; next it uses elementary transformations to reduce a matrix to staircase form, where the number of nonzero rows is the rank of the matrix — a quantity that measures how many independent dimensions of information the matrix carries.

After reading this chapter you should be able to explain: why matrix multiplication is not commutative? why the rank describes the amount of information in a matrix? These questions will recur throughout the later chapters.