Answer to Question #42310 in Linear Algebra for Anurodh

Question

a) Let V ={(a,b,c,d)∈R^4|a + b +2c + 2d = 0} and W ={(a,b,c,d)∈R^4|a= −b,c= −d}. Check that V and W are vector spaces. Further, check that W is a subspace of V.

b) Find the dimensions of V and W.

c) Let P^3 ={ax^3+bx^2+cx+d| a,b,c,d∈R}. Check whether f(x) = x^2+2x+1 is in [S], the subspace of P^3 generated by S ={3x^2+1, 2x^2+x+1}. If f(x) is in [S], write f as a linear combination of elements in S. If f(x) is not in [S], find another polynomial g(x) of degree at most two such that f(x) is in the span of S∪{g(x)}. Also write f as a linear combination of elements in S∪{g(x)}.