Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Notation and convention
- 1 Vector spaces
- 2 Linear mappings
- 3 Determinants
- 4 Scalar products
- 5 Real quadratic forms and self-adjoint mappings
- 6 Complex quadratic forms and self-adjoint mappings
- 7 Jordan decomposition
- 8 Selected topics
- 9 Excursion: Quantum mechanics in a nutshell
- Solutions to selected exercises
- Bibliographic notes
- References
- Index
3 - Determinants
Published online by Cambridge University Press: 18 December 2014
- Frontmatter
- Dedication
- Contents
- Preface
- Notation and convention
- 1 Vector spaces
- 2 Linear mappings
- 3 Determinants
- 4 Scalar products
- 5 Real quadratic forms and self-adjoint mappings
- 6 Complex quadratic forms and self-adjoint mappings
- 7 Jordan decomposition
- 8 Selected topics
- 9 Excursion: Quantum mechanics in a nutshell
- Solutions to selected exercises
- Bibliographic notes
- References
- Index
Summary
In this chapter we introduce one of the most important computational tools in linear algebra – the determinants. First we discuss some motivational examples. Next we present the definition and basic properties of determinants. Then we study some applications of determinants.
Motivational examples
We now present some examples occurring in geometry, algebra, and topology that use determinants as a natural underlying computational tool.
3.1.1 Area and volume
Let u = (a1, a2) and υ = (b1, b2) be nonzero vectors in ℝ2. We consider the area of the parallelogram formed from using these two vectors as adjacent edges. First we may express u in polar coordinates as
u = (a1, a2) = ∥u∥ (cos θ, sin θ).
Thus, we may easily resolve the vector υ along the direction of u and the direction perpendicular to u as follows
Here c2 may be interpreted as the length of the vector in the resolution that is taken to be perpendicular to u. Hence, from (3.1.2), we can read off the result
c2 = ±(b2 cos θ − b1 sin θ) = |b2 cos θ − b1 sin θ|.
Therefore, using (3.1.3) and then (3.1.1), we see that the area σ of the parallel- ogram under consideration is given by
σ = c2 ∥u∥ = |∥u∥ cos θ b2 − ∥u∥ sin θ b1 | = |a1b2 − a2b1|.
Thus we see that the quantity a1b2 − a2b1 formed from the vectors (a1, a2) and (b1, b2) stands out, that will be called the determinant of the matrix
written as det (A) or denoted by
Since det(A) = ±σ, it is also referred to as the signed area of the parallelogram formed by the vectors (a1, a2) and (b1, b2).
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- A Concise Text on Advanced Linear Algebra , pp. 78 - 114Publisher: Cambridge University PressPrint publication year: 2014