Thread: Can someone explain this linear transformation?

Ok, so the question is as follows: Let V and W be linear spaces, each with dimension 2 and each with basis (e1, e2). Let T:V --> W be a linear transformation such that
T(e1 + e2) = 3e1 + 9e2, T(3e1 + 2e2) = 7e1 + 23e2. Compute T(e2 - e1) and determine the nullity and the rank of T.

So my problem is that I don't understand the two T equation that they give you and how they work. Can anyone lead me to the right direction?

First of all, this looks like an abuse of notation. If V and W truly had the same basis (e_1,e_2), then the vector spaces would be identical. So I suppose the two bases are different, although they have the same name.

So, you are given that T applied to the sum of the two basis vectors of V is equal to 3e1+9e2, where these e1 and e2 constitute a basis of W. This is nothing fancy, think of it as an ordinary function f. You have some x, and you are being told that f(x) is equal to some y. In this case, the function f is T, the element x is e1+e2 and y is 3e1+9e2. The map (or function) T maps the element e1+e2 of V to 3e1+9e2 of W.

Similarly T maps 3e1+2e2 of V to 7e1+23e2 of W.

Given these two pieces of information, you need to figure out what T applied to the vector e2-e1 of V is. For this purpose, notice that

e2-e1 = 5(e1+e2)-2(3e1+2e2).

Now you need to apply that T is linear, which manifests itself in that:

T(e2-e1) = T(5(e1+e2)-2(3e1+2e2)) = 5T(e1+e2)-2T(3e1+2e2).