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**studentmath92** Give and prove an example of a subset A $\displaystyle \subseteq$ $\displaystyle \mathbb{R} ^{2}$ which is closed under scalar multiplication, but not under vector addition.

A = { (x,y) ; |x| - |y| = 0 } . Clearly closed under scalar multiplication but (1, -1) and (1,1) belong to A, but (1,-1) + (1,1) = (2,0) doeen't.

Give and prove an example of a subset A $\displaystyle \subseteq$ $\displaystyle \mathbb{R} ^{2}$ which is closed under vector addition but not under scalar multiplication.