What is the subspace spanned by a single vector in function space?

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The subspace spanned by a single vector in function space, exemplified by the function f(x) = x + 1, consists of all scalar multiples of that vector. Specifically, this means the subspace includes all functions of the form g(x) = a * (x + 1), where 'a' is a scalar from the real numbers (R) or complex numbers (C). This concept is fundamental in understanding the structure of function spaces in linear algebra.

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What would the subspace spanned by a single vector (for example) f(x)=x+1 be?
 
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The subspace spanned by a vector is the set of all scalar multiples of that vector. A function like f(x)=x+1 is a "vector" in some function space, most likely over the real or complex numbers, so the answer is all functions of the form g(x)=a*(x+1), where a is an element of the scalar field (R or C).
 

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