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Real linear subspace closed under multiplication

Source: Czech and Slovak Olympiad 1975, National Round, Problem 6

July 20, 2024
algebralinear algebraSubsets

Problem Statement

Let M⊆R2\mathbf M\subseteq\mathbb R^2 be a set with the following properties: 1) there is a pair (a,b)∈M(a,b)\in\mathbf M such that ab(a−b)≠0,ab(a-b)\neq0, 2) if (x1,y1),(x2,y2)∈M\left(x_1,y_1\right),\left(x_2,y_2\right)\in\mathbf M and c∈Rc\in\mathbb R then also (cx1,cy1),(x1+x2,y1+y2),(x1x2,y1y2)∈M.\left(cx_1,cy_1\right),\left(x_1+x_2,y_1+y_2\right),\left(x_1x_2,y_1y_2\right)\in\mathbf M. Show that in fact M=R2.\mathbf M=\mathbb R^2.
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