Find a basis and dimension of a vector space

[gruba]

Homework Statement

Find basis and dimension of $V,W,V\cap W,V+W$ where $V=\{p\in\mathbb{R_4}(x):p^{'}(0) \wedge p(1)=p(0)=p(-1)\},W=\{p\in\mathbb{R_4}(x):p(1)=0\}$

Homework Equations

-Vector spaces

The Attempt at a Solution

Could someone give a hint how to get general representation of a vector in $V$ and $W$?

 

[Simon Bridge]

Look in your course notes when it talks about "general representation of a vector".
You have definitions of the vector spaces - so start with what the symbols turn into in English.

 

[geoffrey159]

I'm not sure what the operator ^ means in the definition of $V$ so I can't help.
For $W$, you can notice that $W = (X-1) \mathbb{R}_3[X]$ is isomorphic to $\mathbb{R}_3[X]$, so what can you say about the dimension of 2 isomorphic vector spaces ? Secondly, a vector space isomorphism sends a base to a base.

 

[HallsofIvy]

The only thing I have a question about is "p'(0)^p(1)= p(0)= p(-1)". The "^" typically means "and" so one condition is that p(1)= p(0)= p(-1) but the "p'(0)" is incomplete- nothing is said about the derivative at 0. What must be true about it? p in $R^4$ is of the form $ax^4+ bx^3+ cx^2+ dx+ e$ and "p(1)= p(0)= p(-1)" requires that $a+ b+ c+ d+ e= e= a- b+ c- d+ e$. But, again, what is required of "p'(0)= d"?