When we decompose a vector space as a direct sum, the dimensions of the subspaces add to the dimension of the space. The situation with a space that is given as the sum of its subspaces is not as simple. This exercise considers the two-subspace special case.

(a) For these subspaces of M_2×2 find W₁ ∩ W₂, dim(W₁ ∩ W₂), W₁ + W₂, and dim(W₁ + W₂).

(b) Suppose that U and W are subspaces of a vector space. Suppose that the sequence  is a basis for U ∩ W. Finally, suppose that the prior sequence has been expanded to give a sequence  that is a basis for U, and a sequence  that is a basis for W. Prove that this sequence

is a basis for the sum U +W.

(c) Conclude that dim(U +W) = dim(U) + dim(W) - dim(U ∩ W).

(d) Let W₁ and W₂ be eight-dimensional subspaces of a ten-dimensional space. List all values possible for dim(W₁ ∩ W₂).

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