We can define a " left " scalar multiplication operation " : R ?R / I ?
12.
Complex coordinate space is a vector space over the complex numbers, with componentwise addition and scalar multiplication.
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The space of sequences has a natural vector space structure by applying addition and scalar multiplication coordinate by coordinate.
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A geometric interpretation of scalar multiplication is that it stretches, or contracts, vectors by a constant factor.
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This implies that the left and right hand sides are equal up to a non-zero scalar multiplication.
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However, this leads to an extra scalar multiplication in order to choose between-j and + j.
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Thus the addition of displacement vectors expresses the composition of these displacement actions and scalar multiplication as scaling of the distance.
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A ( right-) module homomorphism is a map that is additive ( ) and preserves scalar multiplication ( ).
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This " may not be " a vector space ( it may fail to be closed under scalar multiplication ).
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To " complexify " a space means extending ordinary scalar multiplication of vectors by real numbers to scalar multiplication by complex numbers.
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