Prove that if {X,Y} are a basis of R^2, and U is a 2x2 invertible matrix, then {UX,UY} is also a basis of R^2.

Assume {X,Y} is a basis of R^2 (meaning X and Y are independent) and that U is a 2x2 invertible matrix (det(U)!=0 <=> U is not the 0 matrix). It follows that since X and Y are independent, any linear combination resulting from the non-zero U will remain linearly independent as both are being multiplied by it.