Linear algebra (bases)?

Question

Let S be the subspace of P3 consisting of all polynomials p(x) such that p(0) = 0., and let T be the subspace of all polynomials q(x) suck that q(1) = 0. Find bases for...

S

T

The intersection of S and T.

Answer 1

HOMEWORK! But easy enough that you should see how it's done or you will be in *real* trouble.

If p(x)=ax^3+bx^2+cx+d, the condition p(x)=0 just says that d=0. Letting a,b,c be 1 and the rest 0 in succession gives a base:
x^3, x^2, x for S.

The condition that p(1)=0 means that a+b+c+d=0, so d=-a-b-c. Again, let a,b,c be 1 and the rest 0 in succession to get a base
x^3-1, x^2-1, x-1 for T.

For the intersection, you want *both* d=0 and a+b+c+d=0, so we get d=0 and c=-a-b. Let a and b be 1 and the other 0 to find a base
x^3-x, x^2-x for S intersect T.

Answer 2

those themes are comparable. investigate the define of the vectors in (a). you have (a, 0, 0) = a(one million, 0, 0). So the vectors are scalar multiples of (one million, 0, 0). in case you communicate for a 2d what a beginning place is, an answer might want to return out at you. For (b), play this comparable interest. the climate of the subspace are (a, 3a, 2a) = a(one million, 3, 2).

Answer 3

x^3 0 0
0- x^2 0
0 0 x

(x-1)^3 0 0
0 (x-1)|^2 0
0 0 x-1

intersection product of these basevectors.