2  Abstract Vector Spaces

The step from \(\mathbb{R}^n\) to abstract vector spaces reflects a fundamental principle in mathematics: identifying common patterns to unify seemingly different objects. While \(\mathbb{R}^n\) provides a concrete and visualizable model, the abstract framework reveals that spaces of functions, polynomials, and solutions to some differential equations share the exact same algebraic structure. This abstraction is not just elegant—it’s immensely practical. When we prove a theorem about vector spaces in general, it automatically applies to all these examples at once.

2.1 Definition

A vector space \(V\) over \(\mathbb{R}\) is a set equipped with two operations: vector addition (\(+\)) and scalar multiplication (\(\cdot\)). For \(\mathbf{u},\mathbf{v},\mathbf{w}\in V\) and scalars \(a,b\in\mathbb{R}\), these operations must satisfy:

Vector Addition Properties:

  1. Commutativity: \(\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}\)
  2. Associativity: \((\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})\)
  3. Zero vector: There exists \(\mathbf{0}\in V\) such that \(\mathbf{v}+\mathbf{0}=\mathbf{v}\) for all \(\mathbf{v}\in V\)
  4. Additive inverse: For each \(\mathbf{v}\in V\), there exists \(-\mathbf{v}\in V\) such that \(\mathbf{v}+(-\mathbf{v})=\mathbf{0}\)

Scalar Multiplication Properties:

  1. Distributivity over vector addition: \(a(\mathbf{u}+\mathbf{v})=a\mathbf{u}+a\mathbf{v}\)
  2. Distributivity over scalar addition: \((a+b)\mathbf{v}=a\mathbf{v}+b\mathbf{v}\)
  3. Associativity: \(a(b\mathbf{v})=(ab)\mathbf{v}\)
  4. Identity: \(1\mathbf{v}=\mathbf{v}\)

Examples: Vector spaces appear naturally throughout mathematics and its applications. While we’ll encounter many examples throughout this course, here are a few fundamental ones to illustrate how diverse they can be.

  1. \(\mathbb{R}^n\): Our familiar space of n-tuples of real numbers with the standard addition and scalar multiplication.

  2. Function spaces:

    • \(C[a,b]\): Continuous functions on \([a,b]\)
      • Sum: \((f + g)(x) = f(x) + g(x)\)
      • Scalar multiplication: \((cf)(x) = c\cdot f(x)\)
    • \(C^\infty(\mathbb{R})\): Infinitely differentiable functions
      • Sum: \((f + g)(x) = f(x) + g(x)\)
      • Scalar multiplication: \((cf)(x) = c\cdot f(x)\)

  3. Polynomial spaces:

    • \(\mathbb{P}_n\): Polynomials of degree \(\leq n\)
      • Sum: \((p + q)(x) = p(x) + q(x)\)
      • Scalar multiplication: \((cp)(x) = c\cdot p(x)\)
    • \(\mathbb{P}\): All polynomials
      • Sum: \((p + q)(x) = p(x) + q(x)\)
      • Scalar multiplication: \((cp)(x) = c\cdot p(x)\)

2.2 Subspaces of Vector Spaces

Within any vector space, certain subsets naturally inherit the vector space structure. These special subsets, called subspaces, play a fundamental role in linear algebra.

Definition: A subset \(W\) of the vector space \(V\) is called a subspace if it satisfies three conditions:

  1. The zero vector \(\mathbf{0}\) is in \(W\)
  2. For all \(\mathbf{u},\mathbf{v}\in W\), their sum \(\mathbf{u}+\mathbf{v}\) is also in \(W\) (closed under addition)
  3. For all \(\mathbf{v}\in W\) and all scalars \(c\in\mathbb{R}\), the vector \(c\mathbf{v}\) is in \(W\) (closed under scalar multiplication)

These conditions ensure that \(W\) inherits the vector space structure from \(V\), making it a vector space in its own right and providing us with a rich source of new examples.

Theorem 2.1 Theorem: Every subspace of a vector space \(V\) is itself a vector space.

Proof. Let \(W\) be a subspace of the vector space \(V\). We must verify all eight vector space properties.

Vector Addition Properties:

  1. Commutativity: Let \(\mathbf{u},\mathbf{v}\in W\). Since \(W\subseteq V\), we know \(\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}\), as this holds in \(V\).

  2. Associativity: Let \(\mathbf{u},\mathbf{v},\mathbf{w}\in W\). Since \(W\subseteq V\), we know that \((\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})\), as this holds in \(V\).

  3. Zero vector: This is given directly by subspace property 1.

  4. Additive inverse: Let \(\mathbf{v}\in W\). By subspace property 3, \((-1)\mathbf{v}\in W\). This is the additive inverse of \(\mathbf{v}\) since \(\mathbf{v}+(-1)\mathbf{v}=1\mathbf{v}+(-1)\mathbf{v}=(1-1)\mathbf{v}=0\mathbf{v}=\mathbf{0}\).

Scalar Multiplication Properties:

  1. Distributivity over vector addition: Let \(a\in\mathbb{R}\) and \(\mathbf{u},\mathbf{v}\in W\). We know that \(a(\mathbf{u}+\mathbf{v})=a\mathbf{u}+a\mathbf{v}\), as this holds in \(V\).

  2. Distributivity over scalar addition: Let \(a,b\in\mathbb{R}\) and \(\mathbf{v}\in W\). We know that \((a+b)\mathbf{v}=a\mathbf{v}+b\mathbf{v}\), as this holds in \(V\).

  3. Associativity of scalar multiplication: Let \(a,b\in\mathbb{R}\) and \(\mathbf{v}\in W\). We know that \(a(b\mathbf{v})=(ab)\mathbf{v}\), as this holds in \(V\).

  4. Identity scalar multiplication: Let \(\mathbf{v}\in W\). We know that \(1\mathbf{v}=\mathbf{v}\), as this holds in \(V\), and clearly \(\mathbf{v}\in W\) by assumption.

Therefore, since all eight vector space properties are satisfied, \(W\) is indeed a vector space.

Note that this proof relies heavily on two key facts:

  1. The vector space operations in \(W\) are inherited from \(V\)
  2. The subspace properties ensure that these operations are well-defined on \(W\) (i.e., their outputs remain in \(W\))
Subspaces of \(\mathbb{R}^n\)

Subspaces of \(\mathbb{R}^n\) are fundamental structures that arise naturally in many applications of linear algebra. They have intuitive geometric interpretations that help us visualize abstract concepts:

  1. In \(\mathbb{R}^2\):
    • A line through the origin
    • The entire plane \(\mathbb{R}^2\) itself
    • The zero vector \({\vec{0}}\)
  2. In \(\mathbb{R}^3\):
    • A line through the origin
    • A plane through the origin
    • The entire space \(\mathbb{R}^3\)
    • The zero vector \({\mathbf{0}}\)

In Theorem 1.1, we proved that the span of a set of vectors \({\mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_k}\) in \(\mathbb{R}^n\) is a subspace.

2.3 Inner Products:

An inner product on a vector space \(V\) is a function \(\langle\cdot,\cdot\rangle:V\times V\to\mathbb{R}\) satisfying:

  1. Symmetry: \(\langle\mathbf{u},\mathbf{v}\rangle=\langle\mathbf{v},\mathbf{u}\rangle\)
  2. Linearity: \(\langle a\mathbf{u}+b\mathbf{v},\mathbf{w}\rangle=a\langle\mathbf{u},\mathbf{w}\rangle+b\langle\mathbf{v},\mathbf{w}\rangle\)
  3. Positive definiteness: \(\langle\mathbf{v},\mathbf{v}\rangle\geq 0\) with equality if and only if \(\mathbf{v}=\mathbf{0}\)

The inner product generalizes the familiar dot product of \(\mathbb{R}^n\). Different fields use different notations:

  • In \(\mathbb{R}^n\): \(\mathbf{u}\cdot\mathbf{v}\) (dot product notation)
  • In mathematics: \(\langle \mathbf{u}, \mathbf{v} \rangle\) (angle bracket notation)
  • In physics: \(\langle \mathbf{u} | \mathbf{v} \rangle\) (Dirac or bra-ket notation)

Examples of Inner Products:

  1. Standard dot product in \(\mathbb{R}^n\): \(\langle\mathbf{x},\mathbf{y}\rangle=\sum_{i=1}^n x_iy_i\)
  2. On \(C[a,b]\): \(\langle f,g\rangle=\int_a^b f(x)g(x)\,dx\)
  3. On \(\mathbb{P}_n\): \(\langle p,q\rangle=\int_{-1}^1 p(x)q(x)\,dx\)

Just as in \(\mathbb{R}^n\), these inner products satisfy fundamental properties that make them powerful tools. Every inner product generates a norm through \(\|\mathbf{v}\|=\sqrt{\langle\mathbf{v},\mathbf{v}\rangle}\), which in turn induces a distance function \(d(\mathbf{u},\mathbf{v})=\|\mathbf{u}-\mathbf{v}\|\). The Cauchy-Schwarz inequality holds in any inner product space: \(|\langle\mathbf{u},\mathbf{v}\rangle|\leq\|\mathbf{u}\|\|\mathbf{v}\|\). And the norm satisfies all the properties we know from \(\mathbb{R}^n\): positivity, homogeneity, and the triangle inequality. Thus, every inner product space inherits the geometric structure that makes \(\mathbb{R}^n\) so useful.