Question

hi! Why are the following sets of vectors are not bases for the indicated vector space:
p1 = 1 + x , p2 = 2x-x^2 for P^2
According to my text book: it is because they do not span P^2 . but what does this mean? is there any way to prove this fact?

Answer 1

Any polynomial of degree 2 can be written in the form \[a _{0}+a _{1}x+a _{2}x ^{2}\]
If the given basis elements span Vector Space indicated then any polynomial of degree 2 can be written by the linear combination of p1 and p2
So,
\[\alpha(1+x) +\beta(2x-x ^{2}) = a _{0}+a _{1}x+a _{2}x ^{2} \]
(For all a0,a1,a2 in Real numbers)
Comparing coefficients
\[\alpha+2\beta=a _{1}\]
\[-\beta=a _{2}\]
\[\alpha=a _{0}\]
\[a _{0}-2a _{1}=a _{2}\]
This gives linear dependence relation between a0,a1,a2.
So the given basis is not correct for the indicated vector space.
It can be seen that basis should contain 3 elements as there are three coefficients involved in any polynomial of degree 2.
Basis is degree of freedom in expressing elements of the vector space.

Answer 2

thank you!