Practice Problems for Exam 2 1. Given the matrix A below, using only rational numbers, find the following: (a)The dimension of each fundamental subspace. (b)A “clean”’ basis for each subspace. (c)Which subspaces are orthogonal to each other? Show. A= 3 0 5 −2 6 −9 1 −17 10 −18 15 14 −3 49 25 6 −3 16 −13 7 ~b = −29 148 725 −225 2. Consider the bases B = {(4, 5, −1), (−1, 0, 1), (1, 7, −1)} and B 0 = {(3, 0, −4), (−1, 2, 5), (0, −5, 1)}. (a) Find the matrix representation for PBB0 . The representation should be as the product of two vectors with inverses noted with a power of -1. Do Not multiply out. 0 (b) Find the matrix representation for PBB . The representation should be as the product of two vectors with inverses noted with a power of -1. Do Not multiply out. 3. The eigenvalues of −27 180 310 213 370 A = −30 15 −110 −192 are λ1 = −2, λ2 = 3, and λ3 = −7. (a) Using the RREF form of A − λI, find the eigenvectors for each eigenvalue. Each element of the eigenvectors must be an integer. (b) Construct the matrix P whose columns are the eigenvectors of A such that they are in the same order as the eigenvalues. (c) Without finding P −1 what is P −1 AP ? Explain how you came up with the result. 4. Which of the following sets of polynomials are linearly independent in P 3? (a) {5 − 2x + x2 + 4x3 , 2 + 3x − 6x2 + 7x3 , −1 + 5x2 + 2x3 , 11 − 9x + a9x2 + 9x3 } (b) {5 − 2x + x2 + 4x3 , 2 + 3x − 6x2 + 7x3 , −1 + 5x2 + 2x3 , 11 − 9x + a9x2 + 9x3 } 5. Referring to the previous problem, which sets of polynomials form a bases for P 3 , polynomials of degree less than or equal to 3. 6. (a) Let PB→B 0 be the operator mapping from the basis B to B 0 . Define the operator in terms of matrix multplication. Just indicate an inverse, do not calculate it. (b) Let T be a linear operator that may be defined as T (~x) = A3×3 ~x3×1 where A is define below. Before the mapping T is applied, a vector [~v ]B = [−2, 3, 1]TB is given. Find ~v in the standard basis and apply T . (c) Convert the result in part(b) to the basis B 0 . 7. True or False (a) The set 3, 2 + x2 , 5x − 1, x2 is a basis for P2 . (b) Let V be the vector space of 4×4 skew-symmetric matrices (AT = −A). The dim(V ) = 10. (c) Let V be the vector space of 4 × 4 skew-symmetric matrices. The dim(V ) = 6. " (d) For T : <2 → <2 defined by T x y #! x y #! " x , the set 4x " y , the set 4x = # {T (~e1 ), T (~e2 )} is a basis for <2 . " (e) For T : <2 → <2 defined by T = # {T (~e1 ), T (~e2 )} is a basis for <2 . 8. Assuming that An×n has n distinct eigenvalues, show that for the characteristic polynomial pA (t), that pA (A) = 0n×n . Hint: Why is there a nonsingular matrix S such that S −1 AS = Λ where Λ is a diagonal matrix with the eigenvalues of A on the main diagonal.

Download
# Practice Problems for Exam 2