Find all linear maps F:R^3->R^4 that are onto. Explain.?

Answer 1

There are none. the range of any linear map from R^3 is at most 3 dimensional.

Answer 2

let pj (j = 1..4) be the projaction R4->R1 that takes the jth coordinate. Now F is linear iff pjF is for each j.
Consider the collection of all 4x4 real matrices. a matrix M can be viewed as a map R3->R4 by considering the ith row (a, b, c, d) as the projection onto R1 of the function ax+by+cz+d (where x,y,z are the variables in R3).
It is necessary and sufficient to prove the map induced by a matrix M is onto to show that each of the projections is, which depends only on the requirement that a,b,c are not simultaneously 0.

So the onto maps from R3->R4 are isometric to the set of 4x4 matrices which do not have a row whose first 3 values are 0.