Vector Spaces

Definition

Addition and Scalar multiplication

Given a set \(V\), we define two operations:

• Addition: for every pair of elements \(u, v \in V\), we have \(u + v \in V\)
• Scalar multiplication: for every \(v \in V\) and \(a \in \mathbb{R}\), we have \(av \in V\).

We call elements of \(V\) vectors and elements of \(\mathbb{R}\) scalars. Both addition and scalar multiplication can be thought of as functions. Addition maps two elements in \(V\) to another element in \(V\). Scalar multiplication takes an element in \(V\) and an element in \(\mathbb{R}\) and maps it to an element in \(V\).

Vector space

A vector space is a set \(V\) along with the operations of addition and scalar multiplication that satisfy the following properties. In all these properties \(u, v, w\) are arbitrary vectors in \(V\) and \(a, b\) are scalars in \(\mathbb{R}\):

• commutativity
  • \(u + v = v + u\)
• associativity
  • \((u + v) + w = u + (v + w)\)
  • \((ab)v = a(bv)\)
• additive identity
  • there exists a \(0 \in V\) such that \(v + 0 = v\) for every \(v \in V\)
• additive inverse
  • for every \(v \in V\), there exists a vector \(-v \in V\) such that \(v + (-v) = 0\)
• multiplicative identity
  • \(1v = v\) for all \(v \in V\)
• distributive properties
  • \(a(u + v) = au + av\)
  • \((a + b)v = av + bv\)

Note: The above definition is for a real vector space. When dealing with complex vector spaces, replace \(\mathbb{R}\) with \(\mathbb{C}\).

Examples

• \(\mathbb{R}^{n}\) with the usual addition and scalar multiplication operations
  • \(\mathbb{R}^{2}\) - 2d plane
  • \(\mathbb{R}^3\) - 3d space
• \(\mathbb{M}_{m \times n}(\mathbb{R})\), the set of all real matrices of dimensions \(m \times n\) with the usual rules of addition and scalar multiplication
• \(\mathcal{P}_2(\mathbb{R})\), the set of all polynomials with real coefficients with degree at most \(2\)
• \(\mathcal{F}(X, \mathbb{R})\), the set of all real valued functions on the set \(X\)

References

Linear Algebra Done Right, Sheldon Axler