Skip to content
Free public beta until December 4, 2026: every course is open, no card needed. After that, the first chapter of each course stays free and full access is a one-time $39 ($19 for up to 100 founding members chosen for helpful feedback). Your progress is kept. How the beta works

Recipe

Compute with cosets in a quotient space V/W

Use when: you're working in V/W and need to test coset equality, add or scale cosets, or find \dim(V/W)

Goal: the quotient space's dimension, or a specific coset correctly identified or combined with another

Taught in Quotient spaces V/W

Quick recipe

  1. Find \dim(V/W) from the two known dimensions
  2. Test whether two vectors define the same coset
  3. Add or scale cosets by operating on their representatives
  4. Confirm the answer doesn't depend on which representative was used
  1. 1

    Subtract \dim W from \dim V directly; no basis of either space is needed for this number alone.

    Example

    V=\mathbb R^5 and W is 3-dimensional, so \dim(V/W)=5-3=2.

    • W equals V entirely — \dim(V/W)=0, and V/W collapses to a single point, the coset W itself.
    • W is a proper subspace of V — \dim(V/W) is strictly positive, and distinct cosets genuinely differ.

    Why: a complement of W contributes exactly one new basis vector to V/W per dimension it adds beyond W itself.

  2. 2

    Subtract the two vectors and check whether the difference lies in W; if it does, the two cosets are identical.

    Example

    For W=\operatorname{span}\{(1,0,0)\}: (0,2,1)-(5,2,1)=(-5,0,0)=-5(1,0,0), which lies in W.

    Why: two cosets \mathbf v_1+W and \mathbf v_2+W are equal exactly when \mathbf v_1-\mathbf v_2\in W, since that is when the two translated copies of W coincide entirely.

  3. 3

    Add the two representative vectors together, or multiply one by the scalar, then attach +W to the result.

    Example

    \big[(2,0)+W\big]+\big[(0,3)+W\big] = (2,0)+(0,3)+W = (2,3)+W.

    Why: coset addition is defined exactly this way precisely because it does not depend on which representative was chosen, by Step 2's own equality test.

  4. 4

    Pick a different valid representative of one of the original cosets and check the combined result still names the same final coset.

    Example

    Replacing (2,0) with (3,1)=(2,0)+(1,1) and (0,3) with (-1,2)=(0,3)-(1,1) still gives (3,1)+(-1,2)=(2,3), the same coset.

    Why: this is the well-definedness the technical explanation promises, and this step is proof it actually holds on real numbers.

Common mistakes

  • Testing \mathbf v_1-\mathbf v_2 for membership in the wrong subspace, not W itself.
  • Adding cosets by combining their labels symbolically instead of adding the actual representative vectors.
  • Forgetting \dim(V/W)=0 names a genuine one-element vector space, not "no space at all."
  • Assuming a different choice of representative gives a different coset, instead of checking with Step 2's test.