Final answer:
To show that the space of linear maps L(V, W) is infinite dimensional, one can argue that with a finite basis of V and an infinite dimensional W, there are infinitely many possible linear maps for each basis vector of V, making the set of all such maps uncountable and too large to be generated by any finite set of maps.
Step-by-step explanation:
The student's question is related to the field of linear algebra within mathematics, specifically regarding the dimensionality of spaces of linear mappings. The question asks to prove that if V is a finite dimensional vector space with dim(V) > 0 and W is an infinite dimensional vector space, then the space of linear maps from V to W, denoted as L(V, W), is also infinite dimensional.
To prove this, we can consider that since V is finite dimensional, there is a basis of V with a finite number of vectors. However, since W is infinite dimensional, for each vector in the basis of V, one can construct infinitely many different linear maps, assigning different images in W. Therefore, there cannot be a finite set of linear maps from V to W that can generate all possible linear maps through linear combinations, implying that L(V, W) must be infinite dimensional.