Linear (Vector) Space

Definition. Let $L$ be a set whose elements $x_{1}, x_{2}, \dots$ are of an arbitrary nature, equipped with two operations:

addition: $x + y \in L$ for $x, y \in L$ ,
multiplication by a scalar: $λ x \in L$ for $x \in L, λ \in K$ ,

and these operations satisfy the following axioms of a linear (vector) space over a field $K$ :

A. Closure

$\forall x, y \in L : x + y \in L$
$\forall x \in L, \forall λ \in K : λ x \in L$

B. Linear Axioms

Commutativity of addition: $x + y = y + x$
Associativity of addition: $(x + y) + z = x + (y + z)$
Zero element: $\exists 0 \in L : x + 0 = x$
Opposite element: $\forall x \in L \exists (- x) \in L : x + (- x) = 0$
Distributivity over vectors: $λ (x + y) = λ x + λ y$
Distributivity over scalars: $(λ + μ) x = λ x + μx$
Associativity of scalar multiplication: $λ (μx) = (λ μ) x$
The scalar $1$ does not change a vector: $1 \cdot x = x$

Remarks

The elements $x, y, z \in L$ are called vectors, whatever their actual nature.
The field $K$ determines the type of space: $K = R$ gives a real vector space, $K = C$ gives a complex one. In general one speaks of a vector space over $K$ .

Properties

1. The zero element is unique. Let $0_{1}, 0_{2} \in L$ be two "zero" elements. Then

0_{1} = 0_{1} + 0_{2} = 0_{2}

(the first equality holds because $0_{2}$ is a zero, the second because $0_{1}$ is a zero). Hence $0_{1} = 0_{2}$ .

2. The opposite element is unique, and moreover $(- 1) \cdot x = - x$ .

Examples of Linear Spaces

$V_{1}$ — geometric vectors in the plane; $V_{2}$ — geometric vectors in space.
$R^{n}$ — ordered tuples of $n$ real numbers $(x_{1}, \dots, x_{n})$ with coordinate-wise operations:

x + y = (x_{1} + y_{1}, \dots, x_{n} + y_{n}), λ x = (λ x_{1}, \dots, λ x_{n}) .

This is an $n$ -dimensional linear space over $R$ (similarly $C^{n}$ over $C$ ). 3. $P_{n} (x) = a_{0} + a_{1} x + \dots + a_{n} x^{n}$ — polynomials of degree at most $n$ . The coefficients $(a_{0}, \dots, a_{n})$ behave like vectors in $R^{n + 1}$ , so $dim P_{n} = n + 1$ . 4. The set of all solutions of a homogeneous SLAE $A X = O$ . If $X_{1}, X_{2}$ are solutions and $λ \in R$ , then $A (X_{1} + X_{2}) = O$ and $A (λ X_{1}) = O$ , so the sum and any scalar multiple are again solutions; the remaining axioms are inherited from $R^{n}$ . 5. The set of all matrices of a fixed size $m \times n$ (with entrywise operations). 6. The set of all continuous functions on an interval $C [a; b]$ : the sum of continuous functions and the product with a number are again continuous. This is an infinite-dimensional space.

Sets That Are Not Linear Spaces

The set of polynomials of degree exactly $n$ ( $n \geq 1$ ) is not a linear space: it has no zero element, and it is not closed under addition — the sum of two degree- $n$ polynomials can have a lower degree. For example, with $n = 3$ :

P (x) = x^{3} + 2 x, Q (x) = - x^{3} + 4, P (x) + Q (x) = 2 x + 4

— a degree-1 polynomial. Closure axiom (1) is violated.

Similarly, sets that do not contain the zero vector, or are not closed under the operations, are not linear spaces (for example, vectors in $R^{2}$ with nonnegative coordinates — there are no opposite elements).

The notions of linear dependence of vectors, basis, dimension, and linear span in an arbitrary linear space are treated in the article Basis, Dimension, and Span.