Let f:U-->V and g:V--->W be two linear maps, and let gf:U--->W be the compositio

Question

Let f:U-->V and g:V--->W be two linear maps, and let gf:U--->W be the composition of f and g. (a) If gf is injective then f is injective, but g may not be injective. Give a concrete example using linear maps.
(b) If gf is surjective, then g is surjective, but f may not be surjective. Give a concrete example using linear maps. I know how to do this using arbitrary sets, but I can't think of any examples for linear maps. Help please!
Let f:U-->V and g:V--->W be two linear maps, and let gf:U--->W be the composition of f and g. (a) If gf is injective then f is injective, but g may not be injective. Give a concrete example using linear maps.
(b) If gf is surjective, then g is surjective, but f may not be surjective. Give a concrete example using linear maps. I know how to do this using arbitrary sets, but I can't think of any examples for linear maps. Help please!

Explanation / Answer

You can make your set examples work in the vector space by simply taking the elements of your ground set as a basis for the underlying vector space and the sets in question can be replaced by the vector spaces generated by the sets of vectors from the set example. For instance, (a) U = W = R^2, V = R^3. f(x,y):=(x,y,0). g(x,y,z) = (x,y). Then gf(x,y) = g(x,y,0) = (x,y) so gf is the identity map, but clearly g is not surjective. You can try a similar thing for part b as well.

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