Essentially, you want to show that and are linearly independent if and only if and are linearly independent.
If A = [a b]
c d
is a 2 x 2 matrix such that the vectors [a c] and [b d]
are linearly independent, prove carefully that rank A = 2. (You cannot choose thematrix A - your proof must work for every 2 ⇥ 2 matrix with the property above,i.e. every 2 ⇥ 2 matrix with independent columns.)