Basis, Dimension, and Span

This article continues Linear (Vector) Space: here we introduce linear dependence of vectors, dimension, basis, coordinates, subspaces, and linear spans.

Linear Dependence of Vectors

Definition. Vectors $a_{1}, a_{2}, \dots, a_{k}$ of a linear space $L$ are called linearly dependent if there exist numbers $λ_{1}, \dots, λ_{k}$ , not all zero, such that

λ_{1} a_{1} + λ_{2} a_{2} + \dots + λ_{k} a_{k} = 0 .

If this equality is possible only when $λ_{1} = \dots = λ_{k} = 0$ , the vectors are linearly independent.

Equivalently: vectors are linearly dependent $⟺$ at least one of them is a linear combination of the rest (see Linear Dependence and Independence of Rows for more). In particular, any system containing the zero vector is linearly dependent.

Dimension

Definition. A linear space $L$ is called $n$ -dimensional if it contains $n$ linearly independent vectors, while any $n + 1$ vectors are already linearly dependent. The number $n$ is called the dimension and is denoted $dim L = n$ .

Examples: $dim R^{n} = n$ ; the space of polynomials $dim P_{n} = n + 1$ ; the space of continuous functions $dim C [a; b] = \infty$ (infinite-dimensional).

Basis and Coordinates

Definition. A basis of an $n$ -dimensional space $L$ is any ordered system of $n$ linearly independent vectors $e_{1}, \dots, e_{n}$ .

Theorem. Every vector $x \in L$ has a unique expansion in the basis:

x = x_{1} e_{1} + x_{2} e_{2} + \dots + x_{n} e_{n} .

The numbers $x_{1}, \dots, x_{n}$ are called the coordinates of the vector $x$ in the basis ${e_{i}}$ .

Proof. (uniqueness) Suppose $x = \sum x_{i} e_{i} = \sum x_{i}^{'} e_{i}$ . Subtracting, we get $\sum (x_{i} - x_{i}^{'}) e_{i} = 0$ ; by linear independence of the basis, all $x_{i} - x_{i}^{'} = 0$ , that is, $x_{i} = x_{i}^{'}$ .

Theorem (basis criterion). If $e_{1}, \dots, e_{n}$ are linearly independent and every vector of the space is a linear combination of them, then this is a basis.

Examples of bases: the standard basis $e_{i} = (0, \dots, 1, \dots, 0)$ in $R^{n}$ ; the monomials $1, x, x^{2}, \dots, x^{n}$ in $P_{n}$ . Passing between bases is described by a change-of-basis matrix.

Subspace

Definition. A nonempty subset $L_{1} \subseteq L$ is called a subspace if it is closed under the operations:

$\forall x, y \in L_{1} : x + y \in L_{1}$ ;
$\forall x \in L_{1}, \forall λ \in R : λ x \in L_{1}$ .

Theorem. These two conditions suffice: the remaining axioms of a linear space hold automatically. In particular, $0 = 0 \cdot x \in L_{1}$ and $- x = (- 1) \cdot x \in L_{1}$ . Moreover, $dim L_{1} \leq dim L$ .

Examples of subspaces: the zero subspace ${0}$ ; the solution set of a homogeneous SLAE; any plane or line through the origin in $R^{3}$ .

Linear Span

Definition. The linear span of a system of vectors $a_{1}, \dots, a_{k}$ is the set of all their linear combinations:

V = span (a_{1}, \dots, a_{k}) = {λ_{1} a_{1} + \dots + λ_{k} a_{k} ∣ λ_{i} \in R} .

Theorem. The linear span $V$ is a subspace of $L$ .

Proof. If $x = \sum α_{i} a_{i}$ and $y = \sum β_{i} a_{i}$ belong to $V$ , then $x + y = \sum (α_{i} + β_{i}) a_{i} \in V$ and $λ x = \sum (λ α_{i}) a_{i} \in V$ . Both closure conditions hold.

$V$ is said to be generated by (or spanned by) the vectors $a_{1}, \dots, a_{k}$ . The space itself is the span of its own basis. The dimension of the span equals the rank of the generating system of vectors — the maximum number of linearly independent vectors among them.

Example

In $L = R^{4}$ , find $dim V$ and a basis of the span $V$ generated by

a_{1} = (0; - 1; 2; 0), a_{2} = (1; 0; 2; - 1), a_{3} = (1; - 2; 6; - 1) .

Write the vectors as rows and reduce to row-echelon form:

011 - 1 0 - 2 226 0 - 1 - 1 R_{1} \leftrightarrow R_{2} 101 0 - 1 - 2 226 - 1 0 - 1 R_{3} - R_{1} R_{3} - 2 R_{2} 100 0 - 1 0 220 - 1 00 .

Two nonzero rows remain, so the rank of the system is $2$ .

Answer: $dim V = 2$ ; a basis of the span is, for example, ${(1, 0, 2, - 1), (0, - 1, 2, 0)}$ . The third vector turned out to be redundant: $a_{3} = 2 a_{1} + a_{2}$ .