Let U and W be subspaces of a finite-dimensional vector space V. Prove Grassmanns

Let U and W be subspaces of a finite-dimensional vector space V. Prove Grassmanns Identity: dim(U W) dimU dimW dim(U W) [Hint: The subspace U W is defined in Exercise 48of Section 6.1. Let B {v1,..., vk} be a basis forU W. Extend B to a basis C of U and a basis D of W.Prove that C D is a basis for U W.]