Linear Algebra/Quotient Space

Definition

Let V be a vector space over a field F, and let H be a subspace. Define an equivalence relation where x and y within V are said to be equivalent when x-y is an element of H. Define the sum of two equivalence classes X and Y to be the equivalence containing x+y when x is within X and y is within Y, and the scalar multiple aX where a is an element of F to be the equivalence class containing ax when x is an element of X.

Sums are well-defined because if x₁ and x₂ are within X and y₁ and y₂ are within Y, then x₁+y₁-(x₂+y₂)=(x₁-x₂)+(y₁-y₂) which is an element of H, so their sums are equivalent.

Scalar multiples are also well-defined because if x₁ and x₂ are within X and a is an element of F, then ax₁-ax₂=a(x₁-x₂) which is an element of H, so they are equivalent.

Given equivalence classes X and Y and an element x within X and y within Y and z within Z and an elements a and b within F, X+Y and Y+X both contain x+y and so are the same, (X+Y)+Z and X+(Y+Z) both contain x+y+z and so are the same, H is the identity for addition since any for any h within H, (x+h)-x=h which is within H, so X+H=X, and the equivalence class containing -x is the inverse of X, 1X contains 1x=x and so is the same as X, a(bX) and (ab)X both contain abx and so are the same, and (a+b)X and aX+bX are the same since they both contain ax+bx and a(X+Y) and aX+aY are the same since they both contain ax+ay.

The above paragraph establishes that the equivalence classes with addition and scalar multiplication as define also form a vector space, called the quotient space. This vector space is denoted V/H.

Theorem

If V has dimension d and H has dimension s, then V/H has dimension d-s.