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**ThePerfectHacker** Consider, $\displaystyle \mathbb{R}^{\infty} = \mathbb{R}\times \mathbb{R}\times ... $ - an __infinite __tuple.

Okay, I am abusing notation but I think you have an idea of what I mean.

Now, $\displaystyle \mathbb{R}^{\infty}$ is a vector space over $\displaystyle \mathbb{R}$ in a natural way.

Define, $\displaystyle T: \mathbb{R}^{\infty} \to \mathbb{R}^{\infty}$ by $\displaystyle T(a_1,a_2,a_3,...) = (0,a_1,a_2,a_3,...)$.

Notice that $\displaystyle T(\bold{a}+\bold{b}) = T(\bold{a}) + T(\bold{b})$ and $\displaystyle T(a\bold{a}) = aT(\bold{a})$.

Thus, $\displaystyle T$ is a linear transformation.