Quotient Space - 𝑉/𝑆

Quotient Space - Definition

Let:

• 𝑉 be a vector space over a field 𝐹
• 𝑆 be a subspace of 𝑉

We define an equivalence relation∼ on 𝑉 by stating that 𝑥∼𝑦 iff 𝑥−𝑦 ∈ 𝑆. That is, 𝑥 is related to 𝑦 if one can be obtained from the other by adding an element of 𝑆. From this definition, one can deduce that any element of 𝑆 is related to the zero vector; more precisely, all the vectors in 𝑆 get mapped into the equivalence class of the zero vector.

The equivalence class —or, in this case, the coset— of 𝑥 is often denoted as [𝑥] and is defined as:

• [𝑥] = 𝑥 + 𝑆
• [𝑥] = {𝑥+𝑠 : 𝑠∊𝑆}

The quotient space 𝑉/𝑆 is then often defined as 𝑉/~, the set of all equivalence classes induced by ~ on 𝑉:

• 𝑉/~ = 𝑉/𝑆 = {[𝑥] : 𝑥∊𝑉}
• 𝑉/~ = 𝑉/𝑆 = {𝑥 + 𝑆 : 𝑥∊𝑉}

Scalar multiplication and addition are defined on the equivalence classes by:

These operations turn the quotient space 𝑉/𝑆 into a vector space over 𝐹 with 𝑁 being the zero class ([0] = { 0+𝑠 : 𝑠∊𝑆}).

• i.e. the “zero element” of 𝑉/𝑆 is [0] = { 0+𝑠 : 𝑠∊𝑆}

The mapping that associates 𝑣∊𝑉 to the equivalence class [𝑣] is known as the quotient map 𝜋: 𝑉 → 𝑉/𝑆 defined by:

• 𝜋(𝑣) =  𝑣 + 𝑆

The quotient map 𝜋 is a linear map.

Alternatively phrased, the quotient space 𝑉/𝑆 is the set of all affine subsets of 𝑉 which are parallel to 𝑆

Suppose 𝑆 is a subspace of 𝑉 and 𝑣,𝑤∊𝑉. Then the following are equivalent:

• 𝑣 - 𝑤 ∊ 𝑆
• 𝑣 + 𝑆 = 𝑤 + 𝑆
• (𝑣 + 𝑆) ∩ (𝑤 + 𝑆) ≠ ⦰

Quotient Space - Examples

Lines of a Cartesian Plane

Let:

• 𝑉 = ℝ²
• 𝑆 be a LINE through the origin in 𝑉

Then:

• the quotient space 𝑉/𝑆 is the space/set of all lines in 𝑉 that are parallel to 𝑆

Note that any point of a line will satisfy the equivalence relation because their difference vectors belong to 𝑆

Subspaces of Cartesian Space

Let:

• 𝑉 = ℝ^𝑛
• 𝑆 be a subspace spanned by the first 𝑚 standard basis vectors
  • 𝑆 identified as ℝ^𝑚, consisting of all 𝑛-tuples such that the last 𝑛-𝑚 entries are zero: (𝑣₁, …, 𝑣_𝑚, 0, …, 0)

Then:

• two vectors in ℝ^𝑛are in the same equivalence class modulo the subspace 𝑆 if and only if they are identical in the last 𝑛-𝑚 coordinates.
• the quotient space ℝ^𝑛/ℝ^𝑚 is isomorphic to ℝ^𝑛-𝑚

General Subspaces

More generally, if 𝑉 is an (internal) direct sum of subspaces 𝑈 and 𝑊:

• 𝑉 = 𝑈 ⊕ 𝑊

then the quotient space 𝑉/𝑈 is naturally isomorphic to 𝑊

Quotient Space - Dimension

Suppose 𝑉 is finite-dimensional and 𝑆 is a subspace of 𝑉. Then:

• 𝑑𝑖𝑚(𝑉/𝑆) = 𝑑𝑖𝑚(𝑉) - 𝑑𝑖𝑚(𝑆)

Click here to expand...

• 𝑑𝑖𝑚(𝑉) = 𝑑𝑖𝑚(𝑛𝑢𝑙𝑙-𝑠𝑝𝑎𝑐𝑒 of 𝜋) + 𝑑𝑖𝑚(𝑟𝑎𝑛𝑔𝑒 of 𝜋) # fundamental theorem of linear algebra
• 𝑑𝑖𝑚(𝑉) = 𝑑𝑖𝑚(𝑆) + 𝑑𝑖𝑚(𝑉/𝑆)

Quotient Space - Induced Maps

Suppose 𝑇 is a linear map (i.e. 𝑇∊𝐿(𝑉,𝑊)).

The induced map 𝑇˜: 𝑉/(𝑛𝑢𝑙𝑙 𝑇) → 𝑊 is defined as:

• 𝑇˜(𝑣 + 𝑛𝑢𝑙𝑙 𝑇) = 𝑇𝑣

Induced Map Properties

Suppose 𝑇∊𝐿(𝑉,𝑊). Then:

1. 𝑇˜ is a linear map from 𝑉/(𝑛𝑢𝑙𝑙 𝑇) to 𝑊
2. 𝑇˜ is injective (i.e. every element in the codomain is paired with at most one element in the domain)
3. 𝑟𝑎𝑛𝑔𝑒 𝑇˜ = 𝑟𝑎𝑛𝑔𝑒 𝑇
4. 𝑉/(𝑛𝑢𝑙𝑙 𝑇) is isomorphic to 𝑟𝑎𝑛𝑔𝑒 𝑇 (true because the third statement and that 𝑇˜ is both injective and surjective, thus an isomorphism)

Resources

• YouTube’s Sheldon Axler
• https://en.wikipedia.org/wiki/Quotient_space_(linear_algebra)