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I am a college student who has just started doing Linear Algebra. This is not a homework question, just something abot Linear Transformations that i dont understand. I hope someone can help me. Here it goes:

Consider Vector spaces V and W (over R^4) and a matrix 'A' which maps an element from V to W.

1. Why is it that the basis for the column space for A is exactly the basis for the range space of V in W? [i.e why is dim(col(A))=dim (range(V))]

2. Why is dim (V) = dim (Null space of V) + dim (range space of V in W)?

Thanx in advance :)