Linear Algebra 1 (MA 371), Summer, 1999-2000
Study Guide for Exam 2
You can use Maple to do computations and check your work. The exam will basically cover
4.4, material about determinants from chapter 5 which we discussed in class, 6.1, 6.2, 6.4,
de…nition and properties of positive de…nite matrices in 6.5
I. Computations.
– Gram–Schmidt process
– Factorization: A = QR
– Cramer’s Rule
– Eigensystem (small matrices by hand)
– Factorization: A = SDS ¡1
– Finding projection matrices using A or Q
II. Should know
– algebraic and geometric multiplicities of eigenvalues
– when it’s possible to diagonalize
– spectral mapping theorem
– properties of symmetric matrices
– Spectral theorem (6H)
– properties of positive (semi–) de…nite matrices
– basic properties of determinants
– relationship between det(A); eigenvalues, trace etc.
III. Should be able to give proofs of
– If Q is an orthogonal (square) matrix, then kQxk = kxk ; Qx¢Qy = x¢y; and the eigenvalues
live on the unit circle
– N (A) = N (A> A) for any m £ n matrix
– ¾(Projection matrix) = f0; 1g and …nd eigenspaces of each.
– A> A is symmetric and in fact, positive (semi–)de…nite
– A = SDS ¡1 implies Ak = SDk S ¡1
– Theorem 6D (NOTE: will not be on exam 2 but prepare for …nal)
– Theorem 6J (NOTE: will not be on exam 2 but prepare for …nal)