Introduce the concept of euclidean vector, euclidean space, matrix, and the related notations and terminologies.

The concept of transformation is introduced with many examples from everyday life. Then we think of the left side of a system of linear equations as the formula for a transformation between euclidean spaces. Since such transformations are determined by the coeficient matrices, we call them matrix transformations.

The key property that distinguishes the matrix transformations from the others is the linearity property. In other words, matrix transformations are linear transformations. This viewpoint provides a vivid geometrical picture for matrix transformations (as well as systems of linear equations).

Finally, for transformations between euclidean space, matrix transformations and linear transformations are equivalent. Note that matrix transformations are described by *formula*, while linear transformations are described by *property*. This sets up the groundwork for future possible extension to more abstract setting.

§3 Onto

The connection between linear transformations and systems of linear equations naturally suggests us to revisit the theory of Chapter 1. For a fixed (coefficient) matrix, the range corresponds to all the choices of the right side that makes a system to have solutions. The matrix transformation is onto if the solutions always exist (for all right side).

§4 One-to-one

Continuing the theme of last section, the kernel corresponds to all the freedom involved in the general solution of a system of linear equations. In particulr, a matrix transformation is one-to-one if the solutions is unique.

§5 Composition and Matrix Multiplication

The relation between matrix and linear transformations enables us to compare the operations for the two concepts and gain insights into both concepts. While it is rather straightforward to understand the addition and scalar multiplication of matrices, the multiplication of two matrices is best defined as the operation that corresponds to the composition of linear transformations. The relation is also used to introduce the addition and scalar multiplication for linear transformations.

It is always fruitful to establish relations between different concepts. In order to study one concept, such relations allows one to borrow ideas from the other concept.

§6 Inverse

Continuing the comparision between matrix and linear transformations, the concept of inverse transformation is studied and extended to matrices. Explict formula for the inverse matrix is derived in the 2 by 2 case. Then by relating the inverse to the (unique) solution of systems of linear equations, a way of using row operations to compute the inverse in general is found. Finally the properties of the inverse is discussed.

§7 Transpose

Another natural operation on matrices is introduced. A key function of the transpose is the exchange between the row and column viewpoints of matrices.

For some linear algebra problems, dividing a matrix into meaningful parts is a useful method. The concept of partitioned matrix is introduced and the operations among such matrices are discussed.