Consider two subspaces V and W of R". Show thatdim( V) 4- dim(W) = dim(V fllV) + dim( V
Chapter 3, Problem 67(choose chapter or problem)
Consider two subspaces V and W of R". Show thatdim( V) 4- dim(W) = dim(V fllV) + dim( V + W).(For the definition of V + W, see Exercise 3.2.50.) (Hint: Pick a basis z7|,M2.mm of VOW. Using Exercise 65, construct bases il\, u2........v\. ?2.................of V and wi, u2, . , um, w\, u)2, ..., wq of W.) Show that u\, 2,..., um, 51, V2, , Vp, w\, u>2* , Wq is a basis of V + W. Demonstrating linear independence is somewhat challenging.
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