__Quotient Space__:

If V is a vector space over \( \mathbb{R} \) and W is a subspace of V then \( \frac{V}{W}=\{w+\alpha | \alpha\in V\} \) is quotient space and two operations addition and scalar multiplication on \( \frac{V}{W} \) is defined as follows:

Let \( \alpha, \beta \) be any two arbitrary element of V then \( W+\alpha, W+\beta\in \frac{V}{W} \)

\( (W+\alpha)+(W+\beta)=W+\alpha+\beta \) and

\( c(W+\alpha)=W+c\alpha \), for any \( c\in\mathbb{R} \)

__Theorem__:

**Any two right cosets of \frac{V}{W} are either disjoint or identical.**

__Proof__:

Let \( (W+\alpha) \) and \( (W+\beta) \) be any two right cosets of W in V where \( \alpha, \beta \in V \).

**Claim** : \( (W+\alpha)\cap (W+\beta)=\phi \) or \( (W+\alpha)=(W+\beta) \)

Suppose, if possible, \( (W+\alpha)\cap (W+\beta)\ne\phi \)

Then, we have to prove that \( (W+\alpha)=(W+\beta) \).

As \( (W+\alpha)\cap (W+\beta)\ne\phi \), there exist a vector \( v\in V \) such that \( v\in(W+\alpha)\cap (W+\beta) \).

\( \implies v\in W+\alpha and v\in W+\beta \)

As \( v\in W+\alpha\implies \exists w_1\in W \) such that \( v=w_1+\alpha \)

Similarly, \( v\in W+\beta\implies \exists w_2\in W \) such that \( v=w_2+\beta \)

\( \therefore w_1+\alpha=w_2+\beta \)

\( \implies \alpha-\beta=w_2-w_1 \)

Since,\( w_1, w_2\in W \)and W is a subspace of V, \( \therefore w_2-w_1\in W \).

\( \therefore (\alpha-\beta) \) is also a vector in W. Let \( u=\alpha-\beta \) be a vector in W.

Now, to prove that (i) \( W+\alpha\subseteq W+\beta \)

(ii) \( W+\beta\subseteq W+\alpha \)

Let x be any vector in \( W+\alpha \).

We will prove that \( x\in W+\beta \).

Now, as \( x\in W+\alpha\implies \exists w\in W \) such that,

\( x=W+\alpha \)

\( =w+(\alpha-\beta)+\beta \)

\( =w+u+\beta \)

\( =w’+\beta \)

\( \implies x\in W+\beta \)

\( \therefore W+\alpha\subseteq W+\beta \)

Let \( y\in W+\beta \)

\( \implies y=w’_1+\beta, w’_1\in W \)

\( =w’_1+w”_1+\alpha\in W+\alpha \)

\( \therefore W+\beta\subseteq W+\alpha \)

\( W+\alpha=W+\beta \)

Hence proved.