Linear Algebra 1 (MA 371), Summer, 1999-2000
Study Guide for the Final Exam
Instructions for Part I: 40 questions each worth 1 point – 28 T/F and 12 “always,
sometimes, never”. On Part I, you are not allowed to use anything other than your writing
instrument.
I. Should know
– properties of the four fundamental subspaces of a matrix
– properties of pivots, free variables, rank etc
– LU and QR
– algebraic and geometric multiplicities of eigenvalues
– when it’s possible to diagonalize
– spectral mapping theorem
– properties of symmetric matrices
– Spectral theorem (6H)
– basic properties of determinants
– relationship between det(A); eigenvalues, trace etc.
– properties of orthogonal matrices
– properties of projection matrices
Instructions for Part II. 5 questions each worth 12 points. Be neat and give complete
answers. On Part II, you are allowed to use a clean sheet in maple (however, you may
have any of the maple worksheets in our software directory) and a calculator. You are not
permitted to use anything else on your harddrive.
I. Computations.
– Algebraic and geometric relationship between the four fundamental subspaces
– LU decomposition
– Solving Ax = b using the LU decomposition
– 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
– Spectral Theorem for Symmetric matrices
– SVD
II. 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
– Easy cases of the Spectral mapping theorem.
– Theorem 6D
– Theorem 6J