MATH 110 - PRACTICE FINAL LECTURE 1, SUMMER 2009 August 6, 2009 As on the practice midterm, the ﬁnal will consist of much fewer problems than the number given here — probably six. These problems range from easy to medium to “almost”-hard, but I don’t think any are as diﬃcult as the starred problems from the practice midterm. All vector spaces can be assumed to be nonzero and ﬁnite-dimensional over the ﬁeld of real numbers or the ﬁeld of complex numbers. Have fun! 1. Axler, 6.18 (Hint: By 6.17 it is enough to show that every vector in null P is orthogonal to every vector in range P — use 6.2 to show this) 2. Axler, 6.20 3. Suppose that V is an inner-product space and that U and W are subspaces of V . Show that U ⊆ W if and only if W ⊥ ⊆ U ⊥ . 4. Suppose that S and T are self-adjoint operators on an inner-product space V . Show that ST is self-adjoint if and only if S and T commute. 5.
This is the end of the preview.
access the rest of the document.