Can anyone help me out with this question? Not sure where to start here. HOMEWOR

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Can anyone help me out with this question? Not sure where to start here. HOMEWORK 5, DUE Friday, October 14 Problem 2. Let V and W be subspaces of R". Recall the subspaces V + W and V nW 1 and 2 of homework 4, as well as what you proved about them. Now we want to prove the formula from problems dim( V) + dim(W)--dim(V n W) = dim(V + W). (a) Consider the particular case where the subspaces V and W are two different planes through the origin in R3. What sort of geometric objects are VnW and V +W? What are dim(V), dim(W) dim(VnW), and dim(V +W)?

Explanation / Answer

2)

Let A be a basis of VW. We can extend A to a basis V1 for V and a basis W2 for W. Then, V1 W1 is a basis for V+W, and  V1 W1=A, so

dim(V+W)=|V1 W1 |

  =|V1 |+| W1 |-|V1 W1 |

=|V1 |+| W1 |-|A|

=dim(V)+dim(W)-dim(VW)

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