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I'm reading Shafarevich's Linear Algebra and Geometry. Here he proves the following:

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I am a bit confused: He proves that $F$ is linear iff $(1.8),(1.9)$ are true. But isn't that already called "linear"? It is as if he is showing that if $F$ is linear, then $F$ is linear. If it's of some use to anyone, the proof is in the following links: 1, 2, 3.

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The definition of linear in Shafarevich's Linear Algebra and Geometry is

Definition 1.1 A function $F$ on the set of all rows of length $n$ with values in the set of all numbers is said to be linear if there exist numbers $a_1,a_2,\ldots,a_n$ such that $F$ associates to each row $(c_1,c_2,...,c_n)$ the number (1.7).

where (1.7) is the linear combination: $a_1c_1 + a_2c_2 +\cdots+ a_nc_n$.

So from this starting point the theorem you mention is indeed a theorem and needs proof.

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    Thanks! I guess this was the subtleness I wasn't seeing. – 2017-02-21
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    Probably also the fact that in most linear algebra courses the equations (1.8) and (1.9) above are used as a definition of a linear map and the Definition 1.1 (which is a general definition of linearity) usually follows as a remark. Please accept the answer if you feel it is satisfactory. – 2017-02-21