Linear mapping/Determination on basis/Fact/Proof
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Proof
Since we want and since a linear mapping respects all linear combinations, that is
holds, and since every vector
is such a linear combination, there can at most exist one such a linear mapping.
We define now a
mapping
in the following way: we write every vector with the given basis as
(where for almost all ) and define
Since the representation of as such a
linear combination
is unique, this mapping is well-defined.
Linearity. For two vectors
and ,
we have
The compatibility with scalar multiplication is shown in a similar way, see
exercise.