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.