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.
