Math 415 - Midterm 1Thursday, September 25, 2014Circle your section:Philipp Hieronymi 2pm 3pmArmin Straub 9am 11amName:NetID:UIN:Problem 0. [1 point] Write down the number of your discussion section (for instance, AD2or ADH) and the first name of your TA (Allen, Anton, Babak, Mahmood, Michael, Nathan,Tigran, Travis).Section: TA:To be completed by the grader:0 1 2 3 4 5 6 ShortsP/1 /10 /15 /15 /10 /10 /12 /21 /94Good luck!1Instructions• No notes, personal aids or calculators are permitted.• This exam consists of 9 pages. Take a moment to make sure you have all pages.• You have 75 minutes.• Answer all questions in the space provided. If you require more space to write youranswer, you may continue on the back of the page (make it clear if you do).• Explain your work! Little or no points will be given for a correct answer with noexplanation of how you got it.• In particular, you have to write down all row operations for full credit.Problem 1. LetA =1 0 21 1 01 0 1.(a) [8 points] Determine A−1.(b) [2 points] Check whether AA−1= I3.2Problem 2. Consider the matrixA =3 2 03 1 26 7 5.(a) [10 points] Calculate the LU decomposition of A.(b) [5 points] Solve3 2 03 1 26 7 5x1x2x3=4710without reducing the augmented matrix, but using the LU decomposition.3Problem 3. LetB =1 3 −5 41 4 −8 7−3 −7 9 −6.(a) [8 points] Determine the reduced echelon form of B.(b) [7 points] Use your result in (a) to find a parametic description of the set of solutionsof the following system of linear equations:x1+ 3x2− 5x3= 4x1+ 4x2− 8x3= 7−3x1− 7x2+ 9x3= −64Problem 4. [10 points] Consider the vectorsw =2−4−11, v1=1010, v2=0210, v3=1001.Is w in span{v1, v2, v3}? Show your calculations!5Problem 5. [10 points] Letv1=10−1, v2=0−13, v3=1h1h2.For which values of h1and h2is v3a linear combination of v1and v2?6Problem 6. [12 points] Determine which of the following sets are subspaces and give reasons:(a) W1= {ab: ab = 0},(b) W2= {a + 1a: a in R},(c) W3= {ab: a2+ b2≤ 1}.7SHORT ANSWERS[21 points overall, 3 points each]Instructions: The following problems have a short answer. No reason needs to be given.If the problem is multiple choice, circle the correct answer (there is always exactly one correctanswer).Short Problem 1. Let A be a matrix such that, for everyxyzin R3, Axyz=y + z02x − z.Then, what is A?A =Short Problem 2. Let A =3 0 02 3 00 4 3. Then, what is AT?AT=Short Problem 3. The set of solutions in R3of the equationx1− 3x2+ 2x3= 1is(a) empty,(b) a line not through the origin,(c) a line through the origin,(d) a plane.Short Problem 4. Let A be an l ×m matrix and B be an n×p matrix. Under which conditionis ATB defined?8Short Problem 5. Let C be a 3 × 4 matrix such that C has two pivot columns. Is it true thatthe equation Cx = d has a solution for every d in R3.(a) True.(b) False.(c) Unable to determine.Short Problem 6. LetA =3 a − 63a −a + 6.For which choices of a is the matrix A invertible?Short Problem 7. How many solutions has a linear system with 4 equations and 5 unknowns?(a) The system either has no solution or infinitely many solutions.(b) The system has no solution.(c) The system has exactly one solution.(d) The system has infinitely many
View Full Document
Unlocking...