Quotient Vector Space V/W

At a Glance

• Frequency: 1 sub-part across 1 of 13 years (2025)
• Priority tier: T4
• Marks (count): 8 (1)
• Average solve time: ~12 min
• Difficulty mix: medium 1
• Section: A | Dominant type: proof

Why This Chapter Matters

The quotient space $V/W$ is the natural arena for the First Isomorphism Theorem and the rank–nullity theorem reread through abstract algebra. UPSC 2025 set an 8-mark Section A question on this atom — almost certainly asking for a proof of the dimension formula $\dim(V/W) = \dim V - \dim W$ or a construction of a basis for $V/W$. The topic demands a clean definition-proof style and rewards candidates who can navigate coset notation without getting lost in abstraction.

Minimum Theory

Setup. Let $V$ be a finite-dimensional vector space over a field $\mathbb{F}$, and let $W$ be a subspace of $V$.

Cosets. For $v \in V$, the coset of $v$ modulo $W$ is $v + W = \{v + w : w \in W\} \subseteq V.$

Two cosets are equal iff their representatives differ by an element of $W$: $v_1 + W = v_2 + W \iff v_1 - v_2 \in W.$ The cosets partition $V$.

Quotient space. The set $V/W = \{v + W : v \in V\}$ of all cosets is a vector space under: $\text{Addition: } (v_1 + W) + (v_2 + W) = (v_1 + v_2) + W,$ $\text{Scalar multiplication: } \alpha(v + W) = \alpha v + W.$ These operations are well-defined (independent of the choice of coset representative). The zero element is $0 + W = W$.

Well-definedness check. Suppose $v_1 + W = v_1' + W$, i.e., $v_1 - v_1' \in W$, and similarly $v_2 - v_2' \in W$. Then $(v_1 + v_2) - (v_1' + v_2') = (v_1 - v_1') + (v_2 - v_2') \in W$, so $(v_1+v_2)+W = (v_1'+v_2')+W$. Scalar multiplication is checked similarly. $\square$

Dimension formula (Codimension theorem). $\dim(V/W) = \dim V - \dim W.$

Proof. Let $\dim W = k$ and $\dim V = n$. Choose a basis $\{w_1, \ldots, w_k\}$ for $W$ and extend to a basis $\{w_1, \ldots, w_k, v_{k+1}, \ldots, v_n\}$ for $V$.

Claim: $\mathcal{B} = \{v_{k+1}+W, \ldots, v_n+W\}$ is a basis for $V/W$.

Spanning. Any coset $v + W$ with $v = \sum_{i=1}^k a_i w_i + \sum_{j=k+1}^n b_j v_j$ satisfies $v + W = \sum_{j=k+1}^n b_j v_j + W = \sum_{j=k+1}^n b_j (v_j + W)$ since $\sum_{i=1}^k a_i w_i \in W$ is absorbed into the coset.

Linear independence. Suppose $\sum_{j=k+1}^n c_j (v_j + W) = W$ (the zero coset). Then $\sum_{j=k+1}^n c_j v_j \in W$, so $\sum_{j=k+1}^n c_j v_j = \sum_{i=1}^k d_i w_i$ for some scalars $d_i$. Rearranging, $\sum c_j v_j - \sum d_i w_i = 0$. Since $\{w_1,\ldots,w_k,v_{k+1},\ldots,v_n\}$ is a basis for $V$, all coefficients are zero; in particular $c_j = 0$ for all $j$.

Hence $\mathcal{B}$ is a basis of size $n - k$, giving $\dim(V/W) = n - k = \dim V - \dim W$. $\square$

Analogy. This mirrors the index formula for groups: $|G/N| = |G|/|N|$, with dimension playing the role of logarithm of order.

Question Archetypes

dimension-formula-proof (1 question(s); 2025)

Recognition Cues

• The question asks to “prove” or “establish” the dimension formula.
• May specify $V = \mathbb{R}^n$ or a polynomial space and a concrete $W$; asks for both the proof and an explicit basis.
• 8 marks: the proof itself is the core; a concrete illustration earns the last mark or two.
• Key setup signal: “quotient space $V/W$” or “factor space.”

Solution Template

1. Define $V/W$ and its operations (2–3 lines; do not skip).
2. State what you will prove; choose a basis $\{w_1,\ldots,w_k\}$ for $W$ and extend to $\{w_1,\ldots,w_k,v_{k+1},\ldots,v_n\}$ for $V$.
3. Define $\mathcal{B} = \{v_{k+1}+W,\ldots,v_n+W\}$.
4. Prove spanning: any coset $v+W$ is a linear combination of elements of $\mathcal{B}$ (absorb $W$-components).
5. Prove independence: a zero linear combination forces all coefficients zero via the independence of the basis of $V$.
6. Conclude $\dim(V/W) = n - k$.
7. (If asked) Apply to the given concrete $V$ and $W$.

Worked Example

2025 Paper 1, 2025-P1-Q1d (8 marks)

Let $V = \mathbb{R}^3$ and $W = \{(x, y, z) \in \mathbb{R}^3 : x + y + z = 0\}$. Prove that $V/W$ is a vector space and determine $\dim(V/W)$. Find a basis for $V/W$.

Step 1. Identify $W$ and its dimension.

$W$ is the solution space of $x+y+z=0$, a homogeneous system with one equation in three unknowns. Hence $\dim W = 3 - 1 = 2$.

A basis for $W$: $w_1 = (1,-1,0)$, $w_2 = (1,0,-1)$.

Step 2. Extend to a basis for $V$.

We need to add a vector not in $W$. Take $v_3 = (1,0,0)$; check: $1+0+0 = 1 \neq 0$, so $v_3 \notin W$.

The set $\{w_1, w_2, v_3\} = \{(1,-1,0),(1,0,-1),(1,0,0)\}$ is a basis for $\mathbb{R}^3$ (its determinant equals $-1 \neq 0$). $\checkmark$

Step 3. $V/W$ is a vector space.

$W$ is a subspace of $V$, so the coset operations (addition and scalar multiplication defined above) are well-defined and satisfy all vector space axioms. (The standard verification — closure, associativity, zero element $= W$, negatives — follows directly from the corresponding properties in $V$.) $\square$

Step 4. Basis for $V/W$.

By the proof of the dimension formula, $\mathcal{B} = \{v_3 + W\} = \{(1,0,0)+W\}$ is a basis for $V/W$.

Spanning: Any $(a,b,c) \in \mathbb{R}^3$ can be written as $(a,b,c) = \alpha w_1 + \beta w_2 + \gamma v_3$ for appropriate $\alpha,\beta,\gamma$ (solvable since $\{w_1,w_2,v_3\}$ is a basis). Hence $(a,b,c)+W = \gamma\,v_3 + W = \gamma\,(v_3+W)$.

Independence: If $\gamma(v_3+W) = W$, then $\gamma v_3 \in W$, i.e., $\gamma(1,0,0) \in W$, i.e., $\gamma = 0$.

Step 5. Conclusion.

$\dim(V/W) = \dim V - \dim W = 3 - 2 = 1,$

with basis $\{(1,0,0) + W\}$.

$\boxed{\dim(V/W) = 1, \quad \text{basis: } \{(1,0,0)+W\}.}$

Common Traps

• Confusing the coset $v + W$ with the vector $v$; remember that $v + W = v' + W$ does not imply $v = v'$ — it only implies $v - v' \in W$.
• Skipping the well-definedness check for operations: UPSC proof questions penalise missing steps, even if the conclusion is correct.
• In the spanning argument, forgetting to explain why the $W$-components vanish (they are absorbed into the coset because elements of $W$ shift a coset to itself).
• Giving $\dim(V/W) = \dim V / \dim W$ (division, not subtraction) — a common confusion from the group-index analogy.

Marks-Aware Writing

At 8 marks (Section A, ~12 min), allocate roughly: 1 mark for defining $V/W$ and its operations; 1 mark for the basis-extension setup; 3 marks for the spanning and independence arguments (the core proof); 1 mark for invoking the dimension count; 2 marks for the concrete application (basis, $\dim$). Write the independence argument with a displayed equation — examiners look for the line ”$\sum c_j v_j \in W$, hence $\ldots$, hence $c_j = 0$.”

Practice Set

Only one historical question on this atom (shown above).