MATH 218 Final, Spring 2002

Signature: Student Number: Mathematics 218, Section 3 Final Examination, June 25, 2002, 15:00—-17:00 Important Instructions: 1. This ex consists of two sets of problems. The ten ~“TRUE/FALSE answer” questions are uumbered A.B J. each of them is worth 4 pts. The eight “workout” questions are unmbered 1.2.....8: each of them is worth 7 or 8 pts. The maxim score is 100 points. 2. Do not separate these pages. 3. The pink booklets are merely a somce of scrap paper. [ will not read what is in them. If youn have any commoents/requests write them here: J. Ni . . Nikiel Good Luck! Part I. Write vour answer TRUE 01 FALSE under each of the following problems /statements: ia Problem A. If #7 € R" are not orthogonal. then lal] - ey = le eh. Your answer: Problem B. If Aisa square matrix. then cither the row vectors of A or the column vectors of Ainust be linearly independent. Your answer: Problem C. Lf det(A) = 1. then A is an orthogonal matrix. Your answer: MERION. CNIVER LABRARY OF BELAG? Problem D. If 4 is a matrix of size » x n and E is an elementary matrix of size mn x n, then A and EA must have the same nullspace. Your answer: Problem E. If 4 is a singular matrix. then A must have a row of 0's or a column of 0's. Your answer: Problem fF. Each symunetric matrix can be diagonalized. Your answer: Problem G. If A and B are invertible matrices of the same size, then (4-By~!=A°!. Be}, Your answer: Problem H. If the characteristic polynomial of a matrix A is p(A) = A" — 1, then A is invertible, Your answer: Problem lL. “Pwo planes in 2! can have exactly one common point. Your answer: Problem J. If W is a subspace of R? and MW" is a subspace of V. then V+ is a subspace of Wet, Your answer: Problem -. 7 pts. Let Fo be a subspace of BR” with dim(¥) = 2. Let «© > 0 be a constant. Show that V must contain infinitely many vectors of norm c. nie | Let Yo: Vo AV be a linear transformation. Let ey. rg..... v, € V. Suppose that the Problem 3. 7 pts. vectors Trp). P(g)... Tir;,) ave linearly independent. Prove that then the vectors eyoty..... ey, are linearly independent. some “OO RST ‘RY ictal Problem 2. 7 pts. ee. Let A = la,jjnx9 bean invertible matrix. Write down the formula for A~! in terms of the cofactor expansion. You unist explain all vour notation. eee % vee eS eI Part II. — You inust provide all essential details of vour solution to each of the following problems 8 ou the page that coutains the problem (continue ou the reverse side of that page when needed). Problem |. 7 pts. Prove that if \ is au cigenvalne of a square matrix A= [aj jjix.- then ° is an cigenvalue of A®, Problem 5. 8 pts. Find an orthonormal basis lor the row space of the matrix L | A= il 0 1 2 2 Problem 6. & pts. Find the vector form of the general solution of the following linear system Ar = 6. Then write the vector form of tlhe general solution of Av = 0. Problem 7, & pts. Find a matrix P that diagonalizes the following matrix A. Then find P-'AP.