MATH 415 G83 - WORKSHEET 04. The purpose of this worksheet is to help you
get accustomed to doing linear algebra by hand, not just with Mathematica.
MATH 415 G83 - WORKSHEET 04 The purpose of this worksheet is to help you get accustomed to doing linear algebra by hand, not just with Mathematica . Please note that it is NOT the purpose of the worksheet to explain what you are doing in your homework assignments. You are still expected to read and know the Basics and Tutorials in the courseware.
Notation & Terminology. The following chart shows the dierences in the standard notation
seen in regular textbooks and used by most people, versus that seen in the courseware and Mathematica . Try not to get confused between the two! Standard Mathematica /Courseware singular values SVD stretch factors left-singular vectors SVD hangerframe vectors right-singular vectors SVD alignerframe vectors
Exercise 1 (SVD Analysis). Consider the shear matrix √ A = shear[2 3] =
√ 1 2 3 . 0 1
(a) Find four (right-handed) perpendicular frames such that for each frame {P1 , P2 }, A · P1 and A · P2 are perpendicular to each other. (b) Pick one of the frames from (a). What are the SVD stretch factors of A with respect to this frame? (c) Using the information in parts (a) and (b), write A as a product of a (right-handed) hanger, stretcher, and (right-handed) aligner; that is, nd the singular value decomposition of A. For the following problems, choose the best statement.
Exercise 2 (Multiple Choice). Let A be any non-invertible 2-by-2 matrix. Which of the following is guaranteed to be true? (a) The SVD stretch factors are both positive. (b) At least one of the SVD stretch factors is negative. (c) At least one of the SVD stretch factors is zero. (d) Both of the SVD stretch factors are zero. Exercise 3 (Multiple Choice). Let A be any 2-by-2 matrix. (a) (b) (c) (d) (e) (f)
det(A) = 0 if and only if A is not invertible. det(A) 6= 0 if and only if A is invertible. det(A) = 0 only if one of the SVD stretch factors is negative.
Both (a) and (b) are true. Both (a) and (c) are true. Both (b) and (c) are true.
Exercise 4 (Multiple Choice). Let A be any 2-by-2 matrix with SVD stretch factors x and y. (a) The ellipse resulting from hitting the unit circle with A encloses an area of xyπ .
Date
: February 18, 2013.
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(b) The parallelogram resulting from hitting a square of side length 1 with A encloses an area of x2 y 2 . (c) The absolute value of the determinant of A is xy . (d) Both (a) and (b) are true. (e) Both (a) and (c) are true. (f) Both (b) and (c) are true. (g) (a) through (f) are all true.
Exercise 5 (Multiple Choice). Let A be any 2-by-2 matrix.
(a) If A is invertible, then there is a non-zero vector Y such that A · X = Y has no solution X . (b) If det(A) = 0, then there is a vector X such that A · X = 0. (c) If one of the SVD stretch factors of A is 0, then there is a vector Y such that A · X = Y has in nitely many solutions X . (d) If the result of hitting the unit circle with A is a line segment, then for any non-zero vector Y , there is a unique vector X such that A · X = Y . (e) (a) and (b) are true. (f) (b) and (c) are true. (g) (c) and (d) are true. (h) (a) is false and (e) is true.
