Math 415 - Midterm 2Thursday, October 23, 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 /? /? /? /? /? /? /? /?Good luck!1Instructions• No notes, personal aids or calculators are permitted.• This exam consists of ? 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. Consider the matrixA =1 2 2 11 1 1 10 −1 −1 0.(a) Find a basis for Nul(A).(b) Find a basis for Col(AT).(c) Determine the dimension of Col(A) and the dimension of Nul(AT).Problem 2. Let T : R2→ R3be the linear transformation withT20=222, T0−1=010.Find the matrix A which represents T with respect to the following bases:11,1−1for R2, and200,010,00−1for R3.Problem 3. Consider the matrixA =−1 1 0 0 0−1 0 1 0 00 1 0 −1 00 0 −1 1 00 0 0 −1 10 1 0 0 −1.(a) Draw a directed graph with numbered edges and nodes, whose edge-node incidencematrix is A.(b) Find a basis for Col(AT) by choosing a spanning tree of this graph.(This question is not relevant for the second midterm exam!)(c) Use a property of the graph (briefly explain!) to find a basis for Nul(A).(d) Use a property of the graph (briefly explain!) to find a basis for Nul(AT).2Problem 4. Let V =abcd: a − b = c.(a) Write V as a span.(b) Find a basis for the orthogonal complement of V .Problem 5. Let P2be the vector space of all polynomials of degree up to 2, and let V be thesubspace of polynomials p(t) with the property thatZ20p(t)dt = 0.Find a basis for V . [Hint: Write p(t) = a + bt + ct2and use the integral condition to get acondition on the coefficients of p(t).]Problem 6. Let P3be the vector space of all polynomials of degree up to 3, and let T : P3→ P3be the linear transformation defined byT (p(t)) = tp0(t) − 2p(t).(a) Which matrix A represents T with respect to the standard bases?(b) Find a basis for the null space of A.[Optional but recommended: Interpret in terms of polynomials!]Problem 7. Let a, b be in R. Consider the three vectorsv1=11b, v2=1a0, v3=b1b.(a) For which values of a and b is {v1, v2, v3} a basis of R3?(b) For which values of a and b does span{v1, v2, v3} have dimension 2?Problem 8. Let A =1 2 12 5 0−1 −1 −3.(a) Under which condition(s) on b has the system Ax = b a solution?(b) Find a basis for Nul(A).(c) Note that A1−23=0−8−8. Find all solutions to Ax =0−8−8.3SHORT ANSWERS[?? 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. Give a precise definition of what it means for vectors v1, . . . , vnto belinearly independent.Short Problem 2. Write down a basis for the orthogonal complement of W = span100,110.Short Problem 3. Let A be an 4 × 3 matrix, whose row space has dimension 2. What is thedimension of Nul(A)?Short Problem 4. Let A be an 3 × 3 matrix, whose column space has dimension 3. If b is avector in R3, what can you say about the number of solutions to the equation Ax = b?Short Problem 5. Let A be an 3 × 5 matrix of rank 2. Is it possible to find two linearlyindependent vectors that are orthogonal to the column space of A? For the row space?(a) Possible for both.(b) Possible only for the column space.(c) Possible only for the row space.(d) Not possible in either case.(e) Not enough information to decide.Short Problem 6. Let A be a matrix, and let B be its row reduced echelon form. Which ofthe following is true for any such matrices?(a) Col(A) = Col(B) and Col(AT) = Col(BT)(b) Col(A) = Col(B) and Col(AT) 6= Col(BT)(c) Col(A) 6= Col(B) and Col(AT) = Col(BT)(d) Col(A) 6= Col(B) and Col(AT) 6= Col(BT)(e) None of these are true for all such matrices.Short Problem 7. Let A be a 5 × 4 matrix. Suppose that the linear system Ax = b has thesolution set1 − c + dc3 − 2dd: c, d in R.(a) Give a basis for the null space of A.(b) What is the rank of A?Short Problem 8. The linear system Ax = b has a solution x if and only if . . .(a) b is orthogonal to Col(A).(b) b is orthogonal to Col(AT).(c) b is orthogonal to Nul(A).(d) b is orthogonal to Nul(AT).(e) Neither of these guarantees a solution.Short Problem 9. Let A be the edge-node incidence matrix of a directed graph. Supposethat this graph is not a tree (that is, the graph contains at least one loop). What can you sayabout the rows of
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