This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: MA 35100 GOINS MIDTERM EXAM #2 This exam is to be done in 50 minutes in one continuous sitting. Collaboration is not allowed, but you may use your own personal notes; the lecture notes posted on the course web site; homework solutions posted on the course web site; and Bretschers Linear Algebra with Applications . Write your answers on a separate sheet of paper. Provide all details of your work : either give precise references for or proofs of any statements you claim. Problem 1. Consider the matrix A = 1 1- 1- 1 . a. Compute its reduced row-echelon form. Show all of your work. b. Find a basis for ker( A ). What is the nullity of A ? c. Find a basis for im( A ). What is the rank of A ? Problem 2. Consider a line L in the coordinate plane, running through the origin. Denote proj L : R 2 R 2 as the projection onto L . Explicitly, if ~u = u 1 u 2 is a unit vector parallel to L , then proj L ( ~x ) = A~x in terms of A = u 2 1 u 1 u 2 u 1 u 2 u 2 2 . a. Use geometry to find a basis for im( A ). What is the rank of A ? b. Use the Rank-Nullity Theorem to determine the nullity of A . Give a geometric interpretation of your result. Problem 3. Let B = ( ~v 1 ,~v 2 ) be the basis for R 2 in terms of the vectors ~v 1 = 1 and ~v 2 = 1 . For a scalar k , consider the linear transformation T : R 2 R 2 defined by T ( ~x ) = 1 k 0 1 ~x. Compute the B-matrix of T ....
View Full Document