Please help!
Let $$W = [ \rho(x) \epsilon P _{n}(X) : \rho(0) =0]$$ Find a basis for W

Question

Please help!
Let $$W = [ \rho(x) \epsilon  P _{n}(X)  : \rho(0) =0]$$ Find a basis for W

rock_mit182 · Answer

@ganeshie8

rock_mit182 · Answer

anyone good at Subspaces and basis ?

rock_mit182 · Answer

$$\rho(x)= polynomial $$

rock_mit182 · Answer

@TheSmartOne  @micahm Any ideas ?

rock_mit182 · Answer

@math&ing001  @mathmale  @Cuanchi

math&ing001 · Answer

Since ρ(0)=0, all polinomials belonging to W should look like this:
$ ρ(x)=a_{n}x^n + a_{n-1}x^{n-1} + ..... +a_{1}x$ with $a_{k} \in IR $
So maybe $(x^{n},...,x)$ could be a base for W
But I'm not sure about this, I haven't seen this in ages. Maybe get someone who's still fresh to check on this.

rock_mit182 · Answer

ok thanks... :/

reemii · Answer

What @math&ing001  did is alright.
You want to find $n$? elements of $W$ which are linearly independent and can generate $W$. The proposed base does this.

bobo-i-bo · Answer

$$\{x, x^2,x^3,...,x^n\}$$ is a basis. Basically, $W$ is the set of all polynomial which pass through 0 when x=0, so it is the set of all polynomials with no constant term

thomas5267 · Answer

You can use \in next time.
$$W = [ \rho(x) \in  P _{n}(X)  : \rho(0) =0]$$
Looks much better and syntactically correct.