MHB Linear independence of polynomial set.

bamuelsanks
Messages
3
Reaction score
0
Hi guys,

I've been working on a question which is as follows:

For which real values of c will the set $\{1+cx, 1+cx^2, x-x^2\}$ be a basis for $P_2$?

I'm coming up with the answer as no values of c, but am I really wrong?
I've only checked linear independence, because it would imply that it spans $P_2$ (right?)

I figure one would just create the augmented matrix:
$\left( \begin{array}{ccc} 0 & c & 1 \\ 1 & 0 & 1 \\-1 & 1 & 0\end{array} \right)$

And reduce:
$\left( \begin{array}{ccc} 1 & 0 & \frac{1}{c} \\ 0 & 1 & \frac{1}{c} \\ 0 & 0 & 0\end{array} \right)$

Thanks in advance,
SB
 
Physics news on Phys.org
for any c real number
1+cx = 1+cx^2 + c(x-x^2 )

that means the first one can be written from the other two elements (linear compination from other elements)thats means they can't be a basis.
other way to prove that they are basis or not try to generate the standard basis {1 , x , x^2 } from the given basis if that can be done then they are basis
 
Last edited:
The world of 2\times 2 complex matrices is very colorful. They form a Banach-algebra, they act on spinors, they contain the quaternions, SU(2), su(2), SL(2,\mathbb C), sl(2,\mathbb C). Furthermore, with the determinant as Euclidean or pseudo-Euclidean norm, isu(2) is a 3-dimensional Euclidean space, \mathbb RI\oplus isu(2) is a Minkowski space with signature (1,3), i\mathbb RI\oplus su(2) is a Minkowski space with signature (3,1), SU(2) is the double cover of SO(3), sl(2,\mathbb C) is the...

Similar threads

Replies
3
Views
2K
Replies
2
Views
3K
Replies
4
Views
2K
Replies
5
Views
2K
Replies
4
Views
2K
Replies
1
Views
1K
Replies
2
Views
1K
Back
Top