Exercises 29-34 require knowledge of the sum and direct sum of subspaces, as defined in

Step 1 of 3

(a)

It is given that  and  are finite-dimensional subspaces of a vector space V, then the subspace W1 + W2 is finite-dimensional.

The intersection of   sub spaces  is a subspace of  , and .  Consider the dimension of   is  and the has a basis is .

Because,  is a subspace of , then  the set of linearly independent vector in . Hence the basis  can be written as,

Then the dimension of  as,

Similarly, the basis  can be written as,

Then the dimension of  as,

The above equations implies that,

                                                       ……(1)