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Let $P_k$ denote the set of polynomials of degree at most k and $Q_k = \{ \sum\limits_j c_j p_j(x) q_j(y) \, | \, p_j , q_j \in P_k \} $. I am trying to show that a basis for $Q_k$ is the set $\{x^i y^j \, | \, 0\leq i,j \leq k \} $.

So my idea is to argue that this set consists of $(k+1)^2$ linearly independent elements, where the linear independence comes from the linear independence of the set $\{1,x,...,x^k\}$ which is a basis of $P_k$.

But how can I get that $dimQ_k = (dimP_k)^2=(k+1)^2$ ? It seems reasonable to hold. Any ideas ?

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