Problem 10. Let V be a vector space, let U₁ and U₂ be subspaces of V. Prove that dim(U₁+U₂) = dim(U₁) + dim(U₂) - dim(U₁

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Problem 10 Let V Be A Vector Space Let U And U Be Subspaces Of V Prove That Dim U U Dim U Dim U Dim U 1 (42.87 KiB) Viewed 33 times

Problem 10. Let V be a vector space, let U₁ and U₂ be subspaces of V. Prove that dim(U₁+U₂) = dim(U₁) + dim(U₂) - dim(U₁U₂). Suppose that U₁ and U₂ are finite dimensional and V = U₁+U₂. Using the above, prove that V is the direct sum of U₁ and U₂ if and only if dim(V) = dim(U₁) + dim(U₂). Problem 11. Let V and W be vector spaces over F, let T: V→ W be a linear transformation. Prove that if V is finite-dimensional then dim(V) = dim(ker (T)) + dim(im(T)).