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Span of a set of vectors

The span of a set \(S\) (of vectors) is defined as the set of all finite linear combinations of elements(vectors) of \(S\), and denoted by \(Span(S)\).

\[Span(S) = \{ \sum_{i=1}^n a_i v_i \in V|a_1, a_2, \cdots, a_n \in \mathbb{R}\}\]

Example

  • Let \(S = \{(1,0)\} \in \mathbb{R}^2\)

    • \(Span(S) = \{a(1,0) | a \in \mathbb{R}\} = \{(a,0) | a \in \mathbb{R}\}\)

  • Let \(S = \{(1,1)\} \in \mathbb{R}^2\)

    • \(Span(S) = \{a(1,1) | a \in \mathbb{R}\} = \{(a,a) | a \in \mathbb{R}\}\)

  • Let \(S = \{(1,0,0), (0,1,0)\} \in \mathbb{R}^3\)

    • \(Span(S) = \{a(1,0,0) + b(0,1,0) | a,b \in \mathbb{R}\} = \{(a, b, 0) | a,b \in \mathbb{R}\}\)

Spaning set for a vector space

Let \(V\) be a vector space. A set \(S\) of vectors in \(V\) is called a spanning set for \(V\) if \(V = Span(S)\).

Example

  • If \(S = \{(1,0), (0,1)\}\), then \(Span(S) = \mathbb{R}^2\).
  • If \(S = \{(1,1), (0,1)\}\), then \(Span(S) = \mathbb{R}^2\).
  • If \(S = \{(1,0,0), (0,1,0), (0,0,1)\}\), then \(Span(S) = \mathbb{R}^3\).

Adding vectors to obtain a spanning set for \(\mathbb{R}^3\)

  • Start with \(S_0\) to be the empty set \(\emptyset\).
  • Thus \(Span(S_0) = \{(0,0,0)\}\).
  • We will add the vector \((3,0,0)\) to \(S_0\) to obtain \(S_1\).
  • Now \(Span(S_1) = \{(3,0,0)\}\).
  • Now we will add the vector \((2,2,1)\) to \(S_1\) to obtain \(S_2\).
  • This does not cover the entire vector space \(\mathbb{R}^3\).
  • So we add the vector \((1,3,3)\) to \(S_2\) to obtain \(S_3\).
  • So \(Span(S_3) = \mathbb{R}^3\).
  • \((x,y,z) = \frac{3x-5y+4z}{9}(3,0,0) + (y-z)(2,2,1) + \frac{2z-y}{3}(1,3,3)\)

Basis of a vector space

A basis \(B\) of a vector space \(V\) is linearly independent subset of \(B\) that spans \(V\).

Conditions for a set to be a basis

  • The set \(B\) is linearly independent. \(Span(B) = V\).
  • \(B\) is a maximal linearly independent set.
  • \(B\) is a minimal spanning set.

Finding a basis for a vector space

Method 1

Start with the \(Ø\) and keep appending vectors which are not in the span of the set thus far obtained, until we obtain a spanning set.

Example

  • Let \(V = \mathbb{R}^2\).
  • Let us start with the empty set \(\emptyset\) and append a non-zero vector \((1,2)\) to it.
  • Now choose another vector which is not in the span of the of the earlier vector like \((2,3)\).

Method 2

Take a spanning set and keep deleting vectors which are linear combinations of the other vectors, until the remaining vectors satisfy that they are not a linear combination of the other remaining ones.

Example

  • Let \(V = \mathbb{R}^3\).
  • Let us start with the spanning set \(\{(1,0,0), (1,2,0), (1,0,3), (0,2,3), (0,4,3)\}\).
Span of an Empty Set

The span of an empty set is the zero vector space \(\{0\}\).

Dimension / Rank of a vector space

  • The dimension of a vector space \(V\) is the cardinality of a basis for \(V\).
  • If \(B\) is a basis for \(V\), then rank of \(V\) is the cardinality of \(B\).
  • For every vector space there exists a basis,and all bases of a vector space have the same number of elements (or cardinality); hence, the dimension (or rank) of a vector space (say \(V\)) is uniquely defined and denoted by \(dim(V)\) / \(rank(V)\) respectively.