Question

Let V be a finite dimensional vector space and let U and W be subspaces of V. Prove that V = U (direct sum) W if and only if dim(V)=dim(U) + dim (W) and U intersect W = 0

Answer 1

By definition \((\Rightarrow)\) is obvious.

Answer 2

Let \(\{x_1,\ldots,x_n\}\) be a basis of \(U\) and \(\{y_1,\ldots,y_m\}\) be a basis of \(W\). Show that \(\{x_1,\ldots,x_n,y_1,\ldots,y_m\}\) is a basis of \(V\). From there you can have \(V=U+W\). Then using \(U\cap W=0\) we conclude that \(V=U\oplus W\).

Answer 3

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