Chapter 04: Vector Spaces¶

Prerequisites: Linear Equations (Ch01) · Matrix Algebra (Ch02)

Chapter Outline: Definition of Abstract Vector Spaces → The 8 Axioms → Examples (Coordinate Spaces, Matrix Spaces, Polynomials, Function Spaces) → Subspace Criteria → Linear Combinations and Span → Linear Independence/Dependence → Basis and Dimension → Coordinate Vectors → Four Fundamental Subspaces of a Matrix → Rank-Nullity Theorem → Change of Basis and Transition Matrices

Extension: Vector spaces abstract the concept of "arrows," allowing us to treat physical quantities, signals, and polynomials under a unified framework; it is the stage upon which Linear Transformations (Ch05) act.

If matrices are the skeleton of linear algebra, then vector spaces are its soul. By abstracting specific calculation rules into axioms, we gain a universal method for handling any objects that satisfy the principle of linear superposition. This chapter begins with rigorous definitions and systematically constructs the "scaffold" of a space: basis and dimension.

04.1 Vector Spaces and Axioms¶

Definition 04.1 (Vector Space)

A set $V$ is called a vector space over a field $F$ if it is equipped with addition and scalar multiplication satisfying the 8 axioms: 1. Commutativity and Associativity of Addition. 2. Existence of Zero Vector $\mathbf{0}$. 3. Existence of Additive Inverses. 4. Associativity of Scalar Multiplication. 5. Unit Scalar Identity: $1\mathbf{v} = \mathbf{v}$. 6. Distributivity over Scalars: $c(\mathbf{u}+\mathbf{v}) = c\mathbf{u} + c\mathbf{v}$. 7. Distributivity over Vectors: $(a+b)\mathbf{v} = a\mathbf{v} + b\mathbf{v}$.

04.2 Subspaces and Span¶

Definition 04.2 (Subspace)

A non-empty subset $W$ of $V$ is a subspace if $W$ is closed under addition and scalar multiplication: - If $\mathbf{u}, \mathbf{v} \in W$, then $\mathbf{u} + \mathbf{v} \in W$. - If $\mathbf{u} \in W$ and $c \in F$, then $c\mathbf{u} \in W$.

Definition 04.3 (Span)

The set of all linear combinations of a set of vectors $\{\mathbf{v}_1, \ldots, \mathbf{v}_k\}$ is called their span: $$\operatorname{span}\{\mathbf{v}_1, \ldots, \mathbf{v}_k\} = \{c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k : c_i \in F\}$$

04.3 Basis and Dimension¶

Definition 04.4 (Linear Independence)

A set of vectors is linearly independent if $c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k = \mathbf{0}$ implies $c_i = 0$ for all $i$.

Definition 04.5 (Basis and Dimension)

If a set $B = \{\mathbf{b}_1, \ldots, \mathbf{b}_n\}$ is linearly independent and spans $V$, then $B$ is a basis for $V$. The number of vectors in a basis is the dimension $\dim(V)$.

04.4 Four Fundamental Subspaces¶

Theorem 04.1 (Four Fundamental Subspaces)

For an $m \times n$ matrix $A$: 1. Column Space $C(A)$: Spanned by the columns of $A$, in $\mathbb{R}^m$. 2. Nullspace $N(A)$: All $x$ such that $Ax = \mathbf{0}$, in $\mathbb{R}^n$. 3. Row Space $C(A^T)$: Spanned by the rows of $A$, in $\mathbb{R}^n$. 4. Left Nullspace $N(A^T)$: All $y$ such that $A^T y = \mathbf{0}$, in $\mathbb{R}^m$.

Theorem 04.2 (Rank-Nullity Theorem)

$\dim C(A) + \dim N(A) = n$ (number of columns). That is: Rank + Nullity = Total Columns.

Exercises¶

Solution

Chapter Summary¶

Vector spaces establish the geometric and algebraic foundation of linear algebra:

****: Axiomatization allows us to treat disparate mathematical objects (polynomials, signals) as the same type of entity (vectors).

****: The concepts of basis and dimension quantify the "capacity" and "orientation" of a space, establishing the minimal coordinate system for description.

****: The classification of the four fundamental subspaces reveals the deepest internal construction of a linear operator $A$ and the necessity of dimension conservation between input and output.