Let V be a vector space of dimension n, and let W, W2 be subspaces of...

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Let V be a vector space of dimension n, and let W₁, W2 be subspaces of V. (a) Assume that W₁ W₂ = {0}. Show that dim(W₁ W₂) = dim(W₁) + dim(W₂). ( where the direct sum W₁ W₂ of two subspaces W₁ and W₂ of a vector space V is defined). (b) Assume that dim(W₁) + dim(W₂) = n. Does it follow that W₁ + W₂ = V? Justify your answer by either proving the equality or providing an example where it does not hold. Let V be a vector space of dimension n, and let W₁, W2 be subspaces of V. (a) Assume that W₁ W₂ = {0}. Show that dim(W₁ W₂) = dim(W₁) + dim(W₂). ( where the direct sum W₁ W₂ of two subspaces W₁ and W₂ of a vector space V is defined). (b) Assume that dim(W₁) + dim(W₂) = n. Does it follow that W₁ + W₂ = V? Justify your answer by either proving the equality or providing an example where it does not hold.

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a To show that dimW W dimW dimW when W W 0 we can use the ranknullity theorem By definition W W is t... View the full answer