6: Vector Spaces
- Page ID
- 58868
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this chapter we introduce vector spaces in full generality. The reader will notice some similarity with the discussion of the space \(\mathbb{R}^n\) in Chapter [chap:5]. In fact much of the present material has been developed in that context, and there is some repetition. However, Chapter [chap:6] deals with the notion of an abstract vector space, a concept that will be new to most readers. It turns out that there are many systems in which a natural addition and scalar multiplication are defined and satisfy the usual rules familiar from \(\mathbb{R}^n\). The study of abstract vector spaces is a way to deal with all these examples simultaneously. The new aspect is that we are dealing with an abstract system in which all we know about the vectors is that they are objects that can be added and multiplied by a scalar and satisfy rules familiar from \(\mathbb{R}^n\).
The novel thing is the abstraction. Getting used to this new conceptual level is facilitated by the work done in Chapter [chap:5]: First, the vector manipulations are familiar, giving the reader more time to become accustomed to the abstract setting; and, second, the mental images developed in the concrete setting of \(\mathbb{R}^n\) serve as an aid to doing many of the exercises in Chapter [chap:6].
The concept of a vector space was first introduced in 1844 by the German mathematician Hermann Grassmann (1809-1877), but his work did not receive the attention it deserved. It was not until 1888 that the Italian mathematician Guiseppe Peano (1858-1932) clarified Grassmann’s work in his book Calcolo Geometrico and gave the vector space axioms in their present form. Vector spaces became established with the work of the Polish mathematician Stephan Banach (1892-1945), and the idea was finally accepted in 1918 when Hermann Weyl (1885-1955) used it in his widely read book Raum-Zeit-Materie (“Space-Time-Matter”), an introduction to the general theory of relativity.
- 6.1: Examples and Basic Properties
- This page provides an overview of vector spaces, outlining their definition, operations, and axioms. Vectors, which can include diverse objects like matrices and polynomials, must satisfy specific properties for vector space qualification. The significance of polynomials and functions as vector spaces is highlighted, alongside concepts of vector subtraction, scalar multiplication, and simplification of expressions. The zero vector space is also defined, confirming it meets vector space criteria.
- 6.2: Subspaces and Spanning Sets
- This page covers the concept of subspaces in vector spaces, detailing the criteria for a subset to qualify as a subspace, including closure under addition and scalar multiplication. It discusses specific examples such as polynomial spaces and differentiable functions as subspaces, while introducing the idea of spans.
- 6.3: Linear Independence and Dimension
- This page covers linear independence, dependence, and the concept of dimension in vector spaces. It defines linear independence in terms of trivial solutions in linear combinations and presents examples, alongside properties of independent sets. The Fundamental Theorem establishes limits on the size of linearly independent sets, while the Invariance Theorem implies that all bases for a vector space have the same number of vectors, defining its dimension.
- 6.4: Finite Dimensional Spaces
- This page covers key concepts in vector space theory, including basis, dimension, and linear independence. It emphasizes that finite-dimensional vector spaces can be constructed from independent subsets and discusses methods to form bases, including examples from polynomial and matrix spaces. The pages also explore properties of vector space sums and direct sums, establishing the relationship between subspaces and their dimensions.
- 6.5: An Application to Polynomials
- This page covers the vector space of polynomials of degree at most \(n\), \(\mathbf{P}_{n}\), highlighting its dimension of \(n + 1\) and basis formed by polynomials of distinct degrees. It introduces key theorems for polynomial factorization and expansion, and discusses Taylor's Theorem for determining polynomial coefficients.
- 6.6: An Application to Differential Equations
- This page covers differentiable functions and differential equations, outlining \(n\)th derivatives and the structure of their solution spaces. It emphasizes that solutions to \(n\)th order equations have dimension \(n\) and discusses characteristic polynomials and basis sets. Additionally, it focuses on second-order equations with complex roots, establishing that functions \(e^{px}\cos(qx)\) and \(e^{px}\sin(qx)\) form a basis for solutions.
- 6.E: Supplementary Exercises for Chapter 6
- This page covers linear independence of functions using the Wronskian determinant, stating that three functions are linearly independent if their Wronskian is nonzero at a point. It also examines an invertible \( n \times n \) matrix \( A \), discussing its implications on the basis of \( \mathbb{R}^n \), as well as the connections between a matrix's column space, rank, and nullity.