11 Let S and T be the subspaces of defined by P3 be and Find

11. Let S and T be the subspaces of defined by P3 be and Find dim(S nT).

Solution

let s and t be the subspaces of p3 be defined by s={p(x)/p(0)=0}

Note that S is all polynomials of the form p(x) = ax2 + bx where a, b are real numbers. This is becuase p(0) = a(0)2 + b(0) = 0 for all a, b. I propose that {x, x2} forms a basis for S. In order to confirm this we must show that the vectors x and x 2 are linearly independent and span S. To show they are linearly independent we must show that: 1(x 2 ) + 2(x) = 0(x 2 ) + 0(x) only has the solution 1 = 2 = 0. Upon grouping the terms we find: 1 = 0 2 = 0 Thus the two vectors are clealry linearly independent. Now to show that the two vectors span S we must show that any element in S which I will represent by p(x) = ax2 + bx can be written as: 1(x 2 ) + 2(x) = ax2 + bx. where 1, 2 are scalars. Upon grouping the terms we find that: 1 = a 2 = b With this solution we have: ax2 + bx = ax2 + bx which means the two vectors span S. 1 Thus, the two vectors are linearly independent and span S which means {x, x2} forms a basis for S.