Let P be the vector space consisting of all polynomials with degree less than or equal to 2, where the field that P is over is the set of rationals. Now let E2 be the subspace of all stuff in P that has a root of 2, let E5 be the subspace of all elements in P that have a root of 5. Show that P=E2+E5, where + is the subspace sum.
This isn't a homework problem, but it's a problem that I've been having difficulty with, so any help would be much appreciated. Thanks
Sep 15th 2010, 11:28 PM
EDIT: sorry, i misread what you meant by subspace sum.
So just to clarify the definitions,
Sep 16th 2010, 06:41 AM
Yeah, that's right.
Sep 16th 2010, 08:29 AM
Any member of E2 is of the form for rational numbers a and b. Any member of E5 is of the form for rational numbers c and d. Any polynomial in the direct sum of E2 and E5 is of the form for rational numbers a, b, c, d, m, and n. Certainly each of the coefficients is a rational number so E2+ E5 is a subspace of P.
Any member of P is of the form with u, v, and w rational numbers. If you can show that, given any rational numbers u, v, and w, there exist rational numbers a, b, c, d, m, and n such that am+nc= u, 2am+ abm+ 5cn+ cdn= v, and 2ab+ 5cd= w, you are done. Since that is 3 equations in 6 numbers, it should not be two difficult to show that there exist at least one solution for every u, v, w. (You should NOT expect that solution to be unique.)