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  3. 4. verify that the cartesian product v × w of two

4. Verify that the Cartesian product V × W of two vector spaces V and W...

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Advance Math

4. Verify that the Cartesian product V × W of two vector spacesV and W over (the same field) F can be endowed with a vector spacestructure over F, namely, (v, w) + (v ? , w? ) := (v + v ? , w + w? ) and c · (v, w) := (cv, cw) for all c ? F, v, v? ? V , and w, w?? W. This “product” vector space (V × W, +, ·) is commonly (andmore appropriately) denoted as V ? W, called the direct sum of Vand W. The Euclidean plane R 2 ? R × R is in fact R ? R. (Remark.This notion of direct sum can be extended to the direct sum offinitely many vector spaces V1 ? V2 ? · · · ? Vk in astraightforward way.)

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