Skip to main content

2. Vector Spaces

Groups and Vector Spaces​

A group consists of a set GG and an operation that combines elements of GG. The operation is closed and associative, has an identity element, and every element has an inverse. If the operation is also commutative, the group is Abelian.

Examples:

  • (Z,+)(\mathbb Z,+) is an Abelian group. Its identity is 00, and the inverse of nn is −n-n.
  • (N0,+)(\mathbb N_0,+) is not a group because its elements do not all have additive inverses in N0\mathbb N_0.
  • (Z,⋅)(\mathbb Z,\cdot) is not a group: although it has the multiplicative identity 11, not every integer has a multiplicative inverse in Z\mathbb Z.

A real vector space has a set VV with vector addition and scalar multiplication:

  • Vector addition: +:V×V→V+:V\times V\to V.
  • Scalar multiplication: ⋅:R×V→V\cdot:\mathbb R\times V\to V.

Under vector addition, the vectors form an Abelian group. Scalar multiplication must also satisfy

λ(x+y)=λx+λy,(λ+ψ)x=λx+ψx,λ(ψx)=(λψ)x,1x=x.\begin{aligned} \lambda(x+y)&=\lambda x+\lambda y,\\ (\lambda+\psi)x&=\lambda x+\psi x,\\ \lambda(\psi x)&=(\lambda\psi)x,\\ 1x&=x. \end{aligned}

The first two laws are distributivity. The third gives associativity between scalar multiplications, and the last says scalar 11 leaves a vector unchanged. The additive group has a zero vector and every vector has an additive inverse. The scalar 00 and zero vector are different kinds of objects, even though 0x0x equals the zero vector.

Examples include Rn\mathbb R^n, where addition and scalar multiplication are componentwise, and the set of real m×nm\times n matrices, where both operations are entrywise.

Test Your Understanding

Questions

  1. What part of a vector-space structure is already an Abelian group, and which operation defines that group?
  2. In (Z,+)(\mathbb Z,+), identify the identity and the inverse of −5-5.
  3. Why isn’t (Z,⋅)(\mathbb Z,\cdot) a group, even though it has an identity?

Answers

  1. The vectors form an Abelian group under vector addition.
  2. The additive identity is the integer 00. The inverse of −5-5 is 55.
  3. Not every integer has a multiplicative inverse in Z\mathbb Z.