Vector Spaces

1.1 Fields

A field \(F\) satisfies the following axioms:

  1. \(\forall a, b \in F\), \(a + b \in F\), \(a \cdot b \in F\).
  2. \(\forall a, b, c \in F\), \((a + b) + c = a + (b + c)\), \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
  3. \(\forall a, b \in F\), \(a + b = b + a\), \(a \cdot b = b \cdot a\).
  4. There exists an element \(0 \in F\) such that \(\forall a \in F\), \(a + 0 = a\).
  5. There exists an element \(1 \in F\), with \(1 \neq 0\), such that \(\forall a \in F\), \(a \cdot 1 = a\).
  6. \(\forall a \in F\), there exists an element \(-a \in F\) such that \(a + (-a) = 0\).
  7. \(\forall a \in F \setminus \{0\}\), there exists an element \(a^{-1} \in F\) such that \(a \cdot a^{-1} = 1\).
  8. \(\forall a, b, c \in F\), \(a \cdot (b + c) = (a \cdot b) + (a \cdot c)\).

Examples

  1. \(\mathbb{Q}\) — the field of rational numbers.

  2. \(\mathbb{R}\) — the field of real numbers.

  3. \(\mathbb{C}\) — the field of complex numbers.

  4. \(\mathbb{F}_p\) — the finite field with \(p\) elements, where \(p\) is a prime number.

1.2 Vector Spaces

A vector space \(V\) over a field \(F\) satisfies the following axioms:

  1. \(\forall x, y \in V\), \(x + y \in V\).
  2. \(\forall a \in F, \forall x \in V\), \(ax \in V\).
  3. \(\forall x, y \in V\), \(x + y = y + x\).
  4. \(\forall x, y, z \in V\), \((x + y) + z = x + (y + z)\).
  5. There exists an element \(0 \in V\) such that \(\forall x \in V\), \(x + 0 = x\).
  6. \(\forall x \in V\), there exists an element \(-x \in V\) such that \(x + (-x) = 0\).
  7. \(\forall x \in V\), \(1x = x\).
  8. \(\forall a,b \in F, \forall x \in V\), \((ab)x = a(bx)\).
  9. \(\forall a \in F, \forall x, y \in V\), \(a(x + y) = ax + ay\).
  10. \(\forall a, b \in F, \forall x \in V\), \((a + b)x = ax + bx\).
  • The elements of the field \(F\) are called scalars and the elements of the vector space \(V\) are called vectors.

Examples

  1. \(F^n\) — the set of all \(n\)-tuples with entries from a field \(F\).

  2. \(\text{M}_{n \times m}(F)\) — the set of all \(n \times m\) matrices with entries from a field \(F\).

  3. \(\text{P}_n(F)\) — all polynomials with coefficients in \(F\) of degree \(\leq n\).

\[ \text{P}_n(F) = \{\, a_0 + a_1x + \dots + a_nx^n \mid a_i \in F \,\} \]

  1. \(\text{P}(F)\) — all polynomials with coefficients in \(F\).

  2. \(\mathcal{F}(S, F)\) — Let \(S\) be any nonempty set and \(F\) be any field, and \(\mathcal{F}(S, F)\) denote the set of all functions from \(S\) to \(F\).

1.3 Subspaces

A subset \(W\) of a vector space \(V\) over a field \(F\) is called a subspace of \(V\) with the operations of addition and scalar multiplication defined on \(V\) satisfies the following axioms:

  1. \(0 \in W\).
  2. \(\forall x, y \in W\), \(x + y \in W\).
  3. \(\forall a \in F, \forall x \in W\), \(ax \in W\).

Theorem 1.1 Any intersection of subspaces of a vector space \(V\) is a subspace of \(V\).

1.4 Linear Combinations

Let \(V\) be a vector space and \(S\) be a nonempty subset of \(V\). A vector \(v \in V\) is called a linear combination of \(S\). if there exist a finite number of vectors \(u_1, u_2, \ldots, u_n \in S\) and scalars \(a_1, a_2, \ldots, a_n \in F\) such that \(v = a_1u_1 + a_2u_2 + \cdots + a_nu_n\).

  • In this case, we also say that \(v\) is a linear combination of \(u_1, u_2, \ldots, u_n\) and call \(a_1, a_2, \ldots, a_n\) the coefficients of the linear combination.

  • The zero vector is a linear combination of any nonempty subset of \(V\).

Let \(S\) be a nonempty subset of a vector space \(V\). The span of \(S\), denoted \(\text{span}(S)\), is the set consisting of all linear combinations of the vectors in \(S\).

  • \(\text{span}(S)\) is a subspace of \(V\).

  • if \(\text{span}(S) = V\), we also say \(S\) generate (or span) \(V\).

1.5 Linear Dependence and Linear Independence

A subset \(S\) of a vector space \(V\) is called linearly dependent if there exist a finite number of distinct vectors \(u_1, u_2, \ldots, u_n\) in \(S\) and scalars \(a_1, a_2, \ldots, a_n\), not all zero, such that \[ a_1u_1 + a_2u_2 + \cdots + a_nu_n = 0. \] A subset \(S\) of a vector space \(V\) is not linearly dependent is called linearly independent.

Theorem 1.2 Let \(V\) be a vector space, and let \(S_1 \subseteq S_2 \subseteq V\). If \(S_2\) is linearly independent, then \(S_1\) is linearly independent.

Theorem 1.3 Let \(S\) be a linearly independent subset of a vector space \(V\), and let \(v\) be a vector in \(V\) that is not in \(S\). Then \(S \cup \{v\}\) is linearly dependent if and only if \(v \in \text{span}(S)\).

1.6 Bases and Dimension

A basis \(\beta\) for a vector space \(V\) is a linearly independent subset of \(V\) that generates \(V\).

Theorem 1.4 Let \(V\) be a vector space and \(\beta = \{u_1, u_2, \ldots, u_n\}\) be a subset of \(V\). Then \(\beta\) is a basis for \(V\) if and only if each \(v \in V\) can be uniquely expressed as a linear combination of vectors of \(\beta\), that is, can be expressed in the form \(v = a_1u_1 + a_2u_2 + \cdots + a_nu_n\) for unique scalars \(a_1, a_2, \ldots, a_n\).

Theorem 1.5 (Replacement Theorem) Let \(V\) be a vector space that is generated by a set \(G\) containing exactly \(n\) vectors, and let \(L\) be a linearly independent subset of \(V\) containing exactly \(m\) vectors. Then \(m \le n\) and there exists a subset \(H\) of \(G\) containing exactly \(n - m\) vectors such that \(L \cup H\) generates \(V\).

Corollary 1.1 Let \(V\) be a vector space having a finite basis. Then every basis for \(V\) contains the same number of vectors.

  1. In \(\text{F}^n\), let \(e_1 = (1, 0, 0, \ldots, 0)\), \(e_2 = (0, 1, 0, \ldots, 0)\), \(\ldots\) ,\(e_n = (0, 0, \ldots, 0, 1)\). \(\{e_1, e_2, \ldots, e_n\}\) is a basis for \(\text{F}^n\) and is called the standard basis for \(\text{F}^n\).

  2. In \(\text{P}_n(\text{F})\) the set \(\{1, x, x^2, \ldots, x^n\}\) is a basis. We call this basis the standard basis for \(\text{P}_n(\text{F})\).

A vector space is called finite-dimensional if it has a basis consisting of a finite number of vectors.

The unique number of vectors in each basis for \(V\) is called the dimension of \(V\) and is denoted by \(\dim(V)\).

A vector space that is not finite-dimensional is called infinite-dimensional.

Examples

  1. The vector space \(\{0\}\) has dimension zero.
  2. The vector space \(\text{F}^n\) has dimension \(n\).
  3. The vector space \(\text{M}_{n \times m}(\text{F})\) has dimension \(nm\).
  4. The vector space \(\text{P}_n(\text{F})\) has dimension \(n + 1\).

Theorem 1.6 Let \(W\) be a subspace of a finite-dimensional vector space \(V\). Then \(W\) is finite-dimensional and \(\dim(W) \le \dim(V)\). Moreover, if \(\dim(W) = \dim(V)\), then \(V = W\).

Theorem 1.7 If \(W\) is a subspace of a finite-dimensional vector space \(V\), then any basis for \(W\) can be extended to a basis for \(V\).