Let $U$ and $W$ be vector spaces. Define a new vector space, consisting of the set $U \times W =$ $\{(\vec{u},\vec{w})| \vec{u} \in U, \vec{w} \in W\}$ along with these operations: $(\vec{u_1},\vec{w_1}) + (\vec{u_2},\vec{w_2}) = (\vec{u_1} + \vec{u_2}, \vec{w_1} + \vec{w_2})$ and $r(\vec{u},\vec{w}) = (r\vec{u},r\vec{w})$.
Find a basis for, and the dimension of, the external direct sum $P_2 \times \Re ^2$ where $P_n$ is the set of polynomials with degree $\leq n$. What is the relationship among $dim(U)$, $dim(W)$, and $dim(U \times W)$?
For the second part of the question are they not all the same dimensions of size 2? if not can someone explain because I don't see any alternative. I also have no clue how to go about finding the basis but I would guess the dimension being asked for in the first half of the questions is also 2.
Hint: A basis for the vector space of all polynomials (with indeterminate $x$ and coefficients in ${\Bbb R}$) of degree $\leq n$ over ${\Bbb R}$ is $\{1,x,\ldots,x^n\}$.
The vector space ${\Bbb R}^2$ over ${\Bbb R}$ has basis $\{(1,0),(0,1)\}$.