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

4. Verify that the Cartesian product V × W of two vector spaces V and W over (the same field) F can be endowed with a vector space structure 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 (and more appropriately) denoted as V ⊕ W, called the direct sum of V and 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 of finitely many vector spaces V1 ⊕ V2 ⊕ · · · ⊕ Vk in a straightforward way.)

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Colby Messinger answered 1 year ago

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