Question

Does this set form a vector space?

Answer 1

it does form a vector space,
do u need to prove it ?

Answer 2

yes, I thought the same. Don't know how to prove it

Answer 3

Clearly if the polynomial \(\large p(t)\) is in vector space, then the polynomial \(\large c*p(t)\) is also in the vector space, yes ?

Answer 4

ofcourse

Answer 5

yep 0 + 0 = 0

Answer 6

why is p(t) always 0

Answer 7

Ah, i made a mistake. p(t) is not 0,
p(6) is 0

Answer 8

ohk and what about q(t)?

Answer 9

would it be q(6)

Answer 10

nope, here is the complete proof :


Say \(p(t)\) and \(q(t)\) exist in vector space, then we show that the polynomial defined by \(r(t) = c_1∗p(t)+c_2∗q(t) \) also exists in vector space

To prove that the polynomial \(r(t)\) exists in the vector space, we need to show that \(r(6) = 0\) :

\(\begin{align} r(6) &= c_1∗p(6)+c_2∗q(6) \\ & = c_1∗0+c_2∗0~~~~~\color{gray}{(\because p(t) , q(t) \in Vector ~space)} \\&= 0 \end{align}\)

So \(r(t)\) also exists in vector space