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HomeMathematicsAlgebraAny two right cosets of V/W are either disjoint or identical

# Quotient Space:

Â  Â  If V is a vector space over â€‹$$\mathbb{R}$$â€‹ and W is a subspace of V then â€‹$$\frac{V}{W}=\{w+\alpha | \alpha\in V\}$$â€‹ is quotient space and two operations addition and scalar multiplication on â€‹$$\frac{V}{W}$$â€‹Â is defined as follows:
Â  Â  Let â€‹$$\alpha, \beta$$â€‹ be any two arbitrary element of V then â€‹$$W+\alpha, W+\beta\in \frac{V}{W}$$â€‹
â€‹$$(W+\alpha)+(W+\beta)=W+\alpha+\beta$$â€‹Â and
â€‹$$c(W+\alpha)=W+c\alpha$$â€‹, for any â€‹$$c\in\mathbb{R}$$â€‹

# Proof:

Let â€‹$$(W+\alpha)$$â€‹ and â€‹$$(W+\beta)$$â€‹ be any two right cosets of W in V where â€‹$$\alpha, \beta \in V$$â€‹.
Claim : â€‹$$(W+\alpha)\cap (W+\beta)=\phi$$â€‹ or â€‹$$(W+\alpha)=(W+\beta)$$â€‹
Suppose, if possible, â€‹$$(W+\alpha)\cap (W+\beta)\ne\phi$$â€‹
Then, we have to prove that â€‹$$(W+\alpha)=(W+\beta)$$â€‹.
As â€‹$$(W+\alpha)\cap (W+\beta)\ne\phi$$â€‹, there exist a vector â€‹$$v\in V$$â€‹ such that â€‹$$v\in(W+\alpha)\cap (W+\beta)$$â€‹.
â€‹$$\implies v\in W+\alpha and v\in W+\beta$$â€‹
As â€‹$$v\in W+\alpha\implies \exists w_1\in W$$â€‹ such that â€‹$$v=w_1+\alpha$$â€‹
Similarly, â€‹$$v\in W+\beta\implies \exists w_2\in W$$â€‹ such that â€‹$$v=w_2+\beta$$â€‹
â€‹$$\therefore w_1+\alpha=w_2+\beta$$â€‹
â€‹$$\implies \alpha-\beta=w_2-w_1$$â€‹
Since,$$w_1, w_2\in W$$and W is a subspace of V, â€‹$$\therefore w_2-w_1\in W$$â€‹.
â€‹$$\therefore (\alpha-\beta)$$â€‹ is also a vector in W. Let â€‹$$u=\alpha-\beta$$â€‹ be a vector in W.
Now, to prove that (i) â€‹$$W+\alpha\subseteq W+\beta$$â€‹
Â Â  Â Â Â  Â Â Â  Â Â Â  Â Â Â  Â Â Â  Â Â Â  Â Â  (ii) â€‹$$W+\beta\subseteq W+\alpha$$â€‹
Let x be any vector in â€‹$$W+\alpha$$â€‹.
We will prove that â€‹$$x\in W+\beta$$â€‹.
Now, as â€‹$$x\in W+\alpha\implies \exists w\in W$$â€‹ such that,
â€‹$$x=W+\alpha$$â€‹
Â  Â  â€‹$$=w+(\alpha-\beta)+\beta$$â€‹
Â  Â  â€‹$$=w+u+\beta$$â€‹
Â  Â  â€‹$$=w’+\beta$$â€‹
â€‹$$\implies x\in W+\beta$$â€‹
â€‹$$\therefore W+\alpha\subseteq W+\beta$$â€‹
Let â€‹$$y\in W+\beta$$
$$\implies y=w’_1+\beta, w’_1\in W$$â€‹
â€‹Â  Â  Â  Â $$=w’_1+w”_1+\alpha\in W+\alpha$$â€‹
â€‹$$\therefore W+\beta\subseteq W+\alpha$$â€‹
â€‹$$W+\alpha=W+\beta$$â€‹
Hence proved.
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