Name 2270-1Introduction to Linear Algebra 2270-1Sample Midterm Exam 2 Fall 2007Exam Date: 31 OctoberInstructions. This exam is designed for 50 minutes. Calculators, books, notes and computers are notallowed.1. (Matrices, determinants and independence) Do two parts.(a) Prove that the pivot columns of A form a basis for im(A).(b) Suppose A and B are both n×m of rank m and rref(A) = rref(B). Prove or give a counterexample:the column spaces of A and B are identical.Start your solution on this page. Please staple together any additional pages for this problem.Name 2270-12. (Kernel and similarity) Do two parts.(a) Illustrate the relation rref(A) = Ek· · · E2E1A by a frame sequence and explicit elementary matricesfor the matrixA =0 1 21 1 02 2 0.(b) Prove or disprove: ker(rref(BA)) = ker(A), for all invertible matrices B.Start your solution on this page. Please staple together any additional pages for this problem.Name 2270-13. (Independence and bases) Do two parts.(a) Let A be a 12 × 15 matrix. Suppose that, for any possible independent set v1, . . . , vk, the set Av1,. . . , Avkis independent. Prove or give a counterexample: ker(A) = {0}.(b) Let V be the vector space of all polynomials c0+ c1x + c2x2under function addition and scalarmultiplication. Prove that 1 − x, 2x, (x − 1)2form a basis of V .Start your solution on this page. Please staple together any additional pages for this problem.Name 2270-14. (Linear transformations) Do two parts.(a) Let L be a line through the origin in R3with unit direction u. Let T be a reflection through L.Define T precisely. Display its representation matrix A, i.e., T (x) = Ax.(b) Let T be a linear transf ormation from Rninto Rm. Let v1, . . . , vnbe the columns of I and let Abe the matrix whose column s are T (v1), . . . , T (vn). Prove that T (x) = Ax.Start your solution on this page. Please staple together any additional pages for this problem.Name 2270-15. (Vector spaces)(a) Show that the set of all 4 × 3 matrices A which have exactly one element equal to 1, and all otherelements zero, form a basis for the vector space of all 4 × 3 matrices.(b) Let S = { a b−a 2b!: a, b real}. Find a basis for S.(c) Let V be the vector space of all functions defined on the real line, using the usual definitions offunction addition an d scalar multiplication. Let S be the set of all polynomials of degree less than 5(e.g., x4∈ V but x5/∈ V ) that have zero constant term. Prove that S is a subspace of V .Start your solution on this page. Please staple together any additional pages for this problem.Extra Problems Exam 2 2270-11. (Matrix facts)(a) Let A be a given matrix. Assume rref (A) = E1E2· · · EkA for some elementary matrices E1, E2,. . . , Ek. Prove that if A is invertible, then A−1is the product of elementary matrices.(b) Suppose A2= 0 for some square matr ix A. Prove that I + 2A is invertible.(c) Pr ove using non-determinant results th at an invertible matrix cannot have two equal rows .(d) Prove using non-determinant results th at an invertible matrix cannot have a row of zeros.(e) Prove that the column positions of leading ones in rref(A) identify independent columns of A. Userref(A) = E1E2· · · EkA from (a) above in your proof details.3. (Kernel properties)(a) Prove or disprove: ker(rref(A)) = ker(A).(b) Prove or disprove: AB = I with A, B possibly non-square implies ker(B) = {0}.(c) Find the best general values of c and d in the inequality c ≤ dim(im(A)) ≤ d. Th e constants dependon the row and column dimensions of A.(d) Prove that similar matrices A and B = S−1AS satisfy nullity(A) = nullity(B).(e) Find a matrix A of size 3 × 3 th at is not similar to a diagonal matrix.4. (Independence and bases)(a) Show that the set of all m × n matrices A which have exactly one element equal to 1, and all otherelements zero, forms a basis for the vector sp ace of all m × n matrices.(b) Let V be the vector space of all polynomials under function addition and scalar multiplication.Prove that 1, x, . . . , xnare independ ent in V .(c) Let A be an n × m matrix. Find a cond ition on A such that each possible set of independent vectorsv1, . . . , vkis mapped by A into independent vectors Av1, . . . , Avk. Prove assertions.(d) Prove that vectors v1, . . . , vkorthonormal in Rnare linearly independent.(e) Let V be the vector space of all polynomials c0+ c1x + c2x2under function addition and scalarmultiplication. Prove that 1 − x, 2x, (x − 1)2form a basis of V .5. (Linear transformations)(a) Let L be a line through the origin in R2with unit direction u. Let T be a reflection through L.Define T precisely. Display its representation matrix A, i.e., T (x) = Ax.(b) Let T be a linear transformation from Rninto Rm. Given a basis v1, . . . , vnof Rn, let A be thematrix wh ose column s are T (v1), . . . , T (vn). Prove that T (x) = Ax.(c) State and prove a theorem about the matrix of representation for a composition of two lineartransformations T1, T2.(d) Define linear isomorphism. Give an example of h ow an isomorphism can be used to find a basis fora subspace S of a vector space V of