MATH 700 Handout #3 - Linear Algebra: Eigenvectors, Quotient Spaces, Commuting Operators -, Study notes of Linear Algebra

Download MATH 700 Handout #3 - Linear Algebra: Eigenvectors, Quotient Spaces, Commuting Operators - and more Study notes Linear Algebra in PDF only on Docsity! MATH 700 Handout #3 Fall, 2008 The starred problems are due on October 23. Throughout V is an n -dimensional vector space. For a different approach than Axler’s (but using material on determinants that you would have to take on faith) see chapters 6 and 7 in the text by K. Hoffman and R. Kunze. 1. * Let A = [ −10 −6 18 11 ] , and let LA : R 2 → R2 be the corresponding linear operator (we are using the standard basis S = 〈e1, e2〉 ). Verify that each of [ 2 −3 ] and [ −1 2 ] is an eigenvector for LA and determine the eigenvalue for each. Find a basis β for R2 so that B = [LA]β is diagonal, and give the matrix P such that B = P−1AP . 2. Recall the construction of V/W and the universal mapping property that it satisfied. a. In this context show that the constructed map T̃ is unique; that is, if T ′ fills in the diagram the same way, so that T ′◦π = T̃ ◦π = T , then T ′ = T̃ . b. Show that (V/W )∗ ∼= W ◦ . This gives yet another way to compute dim(V/W ) . c. * Suppose that W1 and W2 are subspaces of V . Prove that (W1 + W2)/W2 ∼= W1/(W1 ∩ W2) . Hint: please do not try to define maps on equivalence classes, show that these are well-defined and linear, etc. One approach would be to begin with a composition of maps W1 → W1 + W2 → (W1 + W2)/W2 that you understand, and see how you can use the universal mapping property of an appropriate quotient space. 3. * Let V be an n -dimensional vector space over C ; let T and S be linear operators on V . Suppose thsat ST = TS . Show that S and T have an eigenvector in common (not necessarily with the same eigenvalue). As a preliminary step you might want to show that every eigenspace of S contains an eigenvalue for T . It is important in this problem that the field is C .