1. The problem statement, all variables and given/known data prove that dimV=dimU[tex]\bot[/tex]+dimU 2. Relevant equations 3. The attempt at a solution I've done this on paper, and set V=nullT+rangeT where T maps a vector from V to U. Is it safe to assume that nullT and U[tex]\bot[/tex] are the same? Reasoning is that <T(wi), T(uj)>=0 with wi in nullT and uj in U. Since T(wi)=0, wi gets mapped to 0 in U, and given that the dot product=0, wi is orthogonal to uj. Also consider mapping from say the xy plane to x. The y component is not in x, and it is at a right angle to all vectors in the range x itself, so it is orthogonal to x. Same for mapping from x y z to x y. All z components are orthogonal to all x and y components. This isn't the actual proof btw. If I'm wrong here, then I'll try a different approach.