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## Homework Statement

Let W be a subspace of a vector space V, let y be in V and define the set [itex] y + W = \{x \in V | x = y +w, \text{for some } w \in W\} [/itex] Show that [itex]y + W[/itex] is a subspace of V iff [itex]y \in W[/itex].

## Homework Equations

## The Attempt at a Solution

Let W be a subspace of a vector space V, let y be in V and define the set [itex] y + W = \{x \in V | x = y +w,\text{for some } w \in W\} [/itex].

proof(←)

if [itex] y \in W [/itex] then any vector in [itex]x \in y + W[/itex] will satisfy [itex]x=y+w[/itex] such that [itex] y,w \in W [/itex]. Since W is a subspace and is closed under addition, all the vectors in [itex]y+W[/itex] must also be in W, i.e. [itex] y + W = W[/itex].

proof(→)

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I'm stuck here. I'm thinking to do proof by contradiction? [itex] y + W [/itex] is a subspace and [itex] y \not\in W[/itex]. I'm thinking to find a property it violates? But I don't know how to do this. Any hints on how I should proceed?