When we decompose a vector space as a direct sum, the dimensions of the subspaces add to
Question:
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 M2×2 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).
Step by Step Answer:
a The intersection and sum are which have dimensions one and three b We write B U W for the basis fo...View the full answer
