# Linear Algebra/Quotient Space

## Definition[edit]

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 a**x** when **x** is an element of X.

Sums are well-defined because if **x _{1}** and

**x**are within X and

_{2}**y**and

_{1}**y**are within Y, then

_{2}**x**+

_{1}**y**-(

_{1}**x**+

_{2}**y**)=(

_{2}**x**-

_{1}**x**)+(

_{2}**y**-

_{1}**y**) which is an element of H, so their sums are equivalent.

_{2}Scalar multiples are also well-defined because if **x _{1}** and

**x**are within X and a is an element of F, then a

_{2}**x**-a

_{1}**x**=a(

_{2}**x**-

_{1}**x**) which is an element of H, so they are equivalent.

_{2}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 1**x**=**x** and so is the same as X, a(bX) and (ab)X both contain ab**x** and so are the same, and (a+b)X and aX+bX are the same since they both contain a**x**+b**x** and a(X+Y) and aX+aY are the same since they both contain a**x**+a**y**.

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[edit]

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