Sets of linear equations in several variables can be described by their coefficients,
which can be formed into a regular array, called a matrix; the operations used
to solve equations can then be phrased in terms of matrices.
Matrices also have use in describing linear transformations defined on vectors:
these transformations are operations which take vectors into vectors and are
linear in that they take a weighted sum of vectors into the same weighted sum
of what they take the vectors themselves into. Such transformations play an
important role in physics.
The natural operations of addition and composition among transformations (you
compose two transformations by performing one and then the other; you add them
by forming the transformation whose output is the sum of theirs) give rise to
natural definitions for the addition and multiplication of matrices.
We will discuss use of matrices to solve or reduce equations; define the determinant
of a square matrix and describe its evaluation; describe the operations of addition
and multiplication of matrices, and the concepts of inverse, singular matrix,
eigenvalue, and eigenvector, and discuss how these can be found.