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Math 2270 Practice Exam 3 - University of Utah Fall 2012 Name: I€ / Please show all your work, as a worked This is a 50 minute exam. 7 problem is required for full points, and partial credit may be rewarded for some work in the right direction. 1 1. (9 Points) The Four Subspaces • (3 points) The rowspace is the orthogonal complement of the • (3 points) I an n-i x n matrix has a d-dimensional nullspace, then its rowspace has dimension cI — • (3 points) For an m x n matrix A calculate: dim(C(A)) + dirn(N(A )). T 11 2 2. (8 points) Orthogonal Coinplernents complement of the vector space 3 - Find a basis for the orthogonal 3. (20 points) Projections Find the projection of the vector b onto the column space of the matrix A: - / 4 \ —1 I b=( 01I. 1 0 i\ 1 1 1 A=( 011 110) 1 ‘\1) (iIot\ t A / (0] i 0 0 H 1 / 3) I Z 3Z Zl3 10) Y() x1 ) L3ZO 3 \j C O3 I (z 33 I1 I/ 3 ‘5 / IC) P f L( 2 I - (I / f L 31 L3J t[ I Lio H o/ 1 ( 10 I oti) HIc2 - I ( L { I / I )z oI l ijH 31 3 Oo 4. (15 points) Linear Regression Find the least squares regression line through the points (1, 1), (2, 3), and (4. 5). - iH 1 , / 5 A4/I I () At; 43 7 1/3 7 L /L Y I 3+7y [9 (z7 7)((IYL 5 7 7 5. (15 points) Orthonormal Bases Find an orthonormal basis for the vec tor space - / 2 \ \_2) 0 span{/1\ 2),( \2)J p Ii ((70) 33D 10 / lol - -/_ 1 i) H) (0) (1zo)/ 11 - (lLo) (S z [) (o} (z)- (1* (-0 /0) - H)(JI - 5 6. (8 points) Properties of Matrices Using the three fundamental prop erties of the determinant (determinant of the identity is 1, switching rows switches the sign, linearity for each row) prove that a matrix with a repeated row has determinant 0. - A 6- 1IflQ4 y’ -ef(A) 7 7. (15 points) Calculating Determinants the matrix ( - Calculate the determinant of 1 —2 7 9 \ 3 —4 5 5 3 6 1 —1 I 4532) Ye 1-Z7 O / /J77 / (iZ - C C 7C iO 00771cc! )-7 C — IYZ7K icj (ii) 8 /)(2 /7 8. (10 points) Other Ways of Calculating Determinants Calculate the de terminant of the matrix - /4 0 2 (231 1 5 using a cofactor expansion along column 2. 3 j C - 9