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From $\mathbb{R}^2$ to $\mathbb{R}^1$? (like $T(x,y)=a$).

From $\mathbb{R}^m$ to $\mathbb{R}^n$ such that $m>n$?

My intuition says that no such linear transformation exist, because additive and homogeneous wouldn't exist, but I might didn't get the concept of linear transformation correctly. If I'm wrong, can you give an example?

Also, is there like a list of basic Linear Transformations that are common?

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    Zero function? ${}$ – 2017-01-10
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    In fact, the set of linear functions from $\mathbb R^m$ to $\mathbb R^n$ is a $mn$ dimensional vector space. So it is pretty large. – 2017-01-10
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    Ok, is there one with a formula or something? – 2017-01-10
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    Do you know matrices? Any $n\times m$ matrix defines a linear function by $x\mapsto Ax$. So write down any $mn$ numbers and you have a linear function. – 2017-01-10
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    They may not be 1-1, but they are still linear. Projections are a good example. Or, take any linear function $\mathbb R^3\to\mathbb R^3$ and throw away one coordinate. More generally, take any linear function $\mathbb R^{m+k}\to\mathbb R^m$ and throw away $k$ coordinates. – 2017-01-10

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how about $$T:\mathbb{R}^2 \to \mathbb{R}: (x,y) \mapsto x?$$

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    does this formula sets with the property of additive? – 2017-01-10
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    @YaronScherf: it's a good exercise to verify that. – 2017-01-10
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    T(x1,y1)=x1, T(x2,y2)=x2, T(x1+x2,y1+y2)=x1+x2 ? – 2017-01-10