help understanding basis and dimension

I am working on the following two problems:

Consider the subset U = {(x1,4x1-7x2,x2) : x1, x2 are real numbers} of R^3

Assume that U is a subspace of R^3.

Find a basis for the subspace U = {(x1,4x1-7x2,x2) : x1, x2 are real numbers} of R^3 and explain why it is a basis for U.

I have deduced that U has the correct number of elements which is 3.

I need to show that U spans R^3 and that U is linearly independant.

However I am for some reason having a hard time in doing so... I'm pretty sure this is all I need to do? But just can't get this problem started.

Part b) asks to find the dimension of said subspace. The dimension (according to my notes) of R^n with standard operations is n. So in this case it is 3 right ?