gagsg Write as a linear combination of or state that it is n

Write as a linear combination of or state that it is not possible. Show that the set of all second degree polynomials with standard operations is not a vector space by showing each of the vector space axioms that fail.

Solution

1.

If v is linear combination of other vectors then it can be written as

v = au₁ + bu₂ +c u₃

= <5a, -6a, -1a> +<2b, -1b, 2b> +<1c, -3c, -2c>

= <5a+2b+c, -6a-b-3c, -a+2b-2c>

=<-5, 8, 7>

Therefore, we can write following simultaneous equations

5a+2b+c = -5

-6a-b-3c = 8

-a+2b-2c = 7

Solving these simultaneous equations we get

a= -3; b= 4 and c= 2

Therefore, v can be written as linear combination of other vectors as v = -3u₁ + 4u₂ + 2u₃

2.

One of the axiom of vector space is Closure under addition

That is if u and v is in vector space V then u+v should be in V.

But we can give counter example like

u = 2x^2+x+1

v = -2x^2+2x+2

u + v = 3x +3

Here, u and v are second degree polynomial but u+v is first degree polynomial so it not in same vector space.

Thus, set of all second degree polynomials is not a vector space.