A linear transformation is a structure-preserving function of one vector space to another vector space over the same field. The structure that must be preserved is that of vector addition and scalar multiplication of which the geometric analogues, when the field is the real numbers, are parallelograms with one vertex at the origin (for vector addition) and lines through the origin (for scalar multiplication). In particular for every linear transformation the image of the zero vector is the zero vector. In this chapter we show that if the linear transformation is from Vm(F) to Vn(F), then it may be represented by multiplication by a unique m × n matrix. The rows of this matrix are the images of the basis vectors. Conversely, multiplication by any m × n matrix gives a linear transformation from Vm(F) to Vn(F).

Concurrent reading: Birkhoff and MacLane, chapter 8, section 1.

1. 1 Which of the functions of qn 11.14 preserve scalar multiplication in the sense that if v ↦ v′ then av ↦ av′ for all a ∈ ℝ?

2. 2 A function Vm(F) → Vn(F) which preserves the structure of vector addition and the structure of scalar multiplication, in the sense that if

3. v ↦ v′ and u ↦ u′ then v + u ↦ v′ + u′ and av ↦ av′

4. for all a ∈ F, is called a linear transformation of Vm(F).) What is the image of the zero vector under any linear transformation?

5. 3 […]