Linear algebra: Find basis and dimension of a vector space

[complier]

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

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

 

[Ishuda]

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

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

I believe you mean the following: The space $\displaystyle \mathbb{R_4}(x)$ can be modeled as the polynomial
p(x) = a x³ + b x² + c x + d
or as the vector space
$\displaystyle \boldsymbol{U}$ = {u = <a, b, c, d>; a, b, c, and d $\displaystyle \epsilon\, \mathbb{R}$}.
Depending on just what is meant by $\displaystyle p^{'}(0) \,\wedge p(1)$ one interpretation or the other might be more useful.