Applied Mathematical Methods

3

Operational Fundamentals of Linear Algebra

In this chapter, we develop the operational fundamentals of linear algebra, which will be used throughout the text in appropriate contexts, especially in the initial chapters. To that end, we analyze the range space and null space of a linear transformation with reference to its matrix. Next, we study the crucial concept of basis and develop the concept and procedure for the change of basis. Finally, as a prelude to our study of systems of linear equations, we establish the ideas of elementary transformations.

Range and Null Space: Rank and Nullity

Consider a real m × n matrix A, denoted as A ∈ R^m×n. As discussed in the previous chapter, it maps a vector x ∈ Rⁿ to a vector y ∈ R^m according to Ax = ...