For which values of a b c d e f are the vectors a 0 0 b c 0

For which values of a, b, c, d, e, f are the vectors [a 0 0], [b c 0], [d e f] a basis for R^3? Explain.

Solution

Let A be as in part (a). Then A ker f if and only if a + d = 0 if and only if a = d. Thus, A ker f if and only if A = a b c a = a 1 0 0 1 + b 0 1 0 0 + c 0 0 1 0 . Since the three matrices above span ker f and are clearly linearly independent, they form a basis for ker f. It is easy to see that f is onto. (If a R then f( a 0 0 0 ) = a.) A basis for im f is therefore {1}