Suppose U and W are subspaces of a vector space V Suppose th

Suppose U and W are subspaces of a vector space V. Suppose that {u_1,. u_n}, and {w_1,. w_m} are bases of U and W respectively, and that U n W = {0}. Prove that the set {u_1,. u_n, w_1,. w_m} is a basis for U + W. Show by example that if {u_1,. u_n}, and {w_1,. w_m} are bases of U and W respectively, and if U n W is not equal to {0}, then the set {u_1,. u_n, w_1,. w_m} need not be a basis for U + W. Be sure to explicitly define U , W and V and show your example satisfies the hypothesis and conclusion.

Solution

a) Given U and W are subspaces are independent of each other

We know that intersection is defined as those elements present in both the sets but from given, there are clearly no common elements. So the intersection of U and W is {0}

So when we add the subspaces, we include all the terms in U and all the elements in W. As there is no common elements, we are adding U and W. Hence it would be U+W