Suppose W₁ and W₂ are finite dimensional subspaces of a vector space V so that the only vector in both subspaces is the zero vector, that is W₁ W₂ = {0}. Show that dim(W₁) + dim(W₂) by taking a basis v₁, . . . , Vn for W₁ and a basis u₁, ..., um for W₂ and arguing that v₁, ..., Vn, U1, . . ., Um is a basis for W₁ + W2, that is v₁, . . . , Vn, U1, . . ., um is both linearly independent and spans W₁ + W₂. dim(W₁ + W₂) = =

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