Solutions to problems from section 2.8 2. Give a specic reason why the set H R2 displayed below is not a subspace of R2 .
Solution: We can nd two vectors u and v in H with u + v not in H . For example:
4. Give a specic reason why the set H R2 displayed be
Solutions to problems from section 1.9 For questions 2, 4, 10 assume that T is a linear map and nd the standard matrix A for T . [Recall that by theorem 10 we know the columns of A will be the images of the ei s under T .] 2. T : R3 R2 , T (e1 ) = Solutio
Solutions to problems from section 2.1 Throughout this assignment you were to assume the each matrix expression is dened. 4. Compute A 5I3 and (5I3 )A when 9 1 3 A = 8 7 6 . 4 1 8 Solution: 4 1 3 9 1 3 500 A 5I3 = 8 2 6 . 7 6 0 5 0 = 8 005 4 1 3 4 1 8 45
Solutions to even problems from section 2.2 2. Find the inverse of the matrix Solution: Since det 32 74 32 74 32 . 74 = 3(4) 7(2) = 2 we have
1
=
1 2
4 2 7 3
=
2 1 7/2 3/2
8. Use matrix algebra to show that if A is invertible and D satises AD = I , then D
Solutions to problems from section 2.3 7 0 4 4. Is the matrix 3 0 1 invertible? 20 9
Solution: NO. The columns of the matrix above are linearly dependent, for 4 7 0 0 0 = 0 3 + 0 + 0 1 . 0 9 0 2
Thus by the inverse matrix theorem (parts (e) and (a) we kno
Solutions to even problems from section 2.4 6. Suppose X0 YZ A0 BC = I0 0I
Find formulas for X, Y , and Z in terms of A, B , and C (assume A and C are square). Solution: Since X0 YZ A0 BC = XA 0 Y A + ZB ZC
we have the following equations XA = I, Y A + ZB
Solutions to even problems from section 2.7 4. Find the 3 3 matrix that produces the following composite 2D transformation using homogeneous coordinates: the y -coordinate by 1.2. 0 2 1 3 whereas the 3 3 0 1 .8 00 matrix for scaling the x-coordinate by .8
Solutions to problems from section 2.9 2. Find a vector x determined by the coordinate vector [x]B = B= Illustrate your answer with a gure. Solution: Let b1 = 2 1 [x]B = Heres a picture: and b2 = 1 3 3 . Then we have 1 x = b1 + 3b2 = 11 2 2 1 , 3 1 . 1 3
Solutions to problems from section 3.1 2. Compute the determinant of the following matrix using the cofactor expansion across the rst row and then compute the determinant by a cofactor expansion down the second column. 0 51 4 3 0 2 41 Solution: 0 51 4 3 0
Solutions to problems from section 1.8 4. With T dened by T (x) = Ax, nd a vector x whose image under T id b and determine whether x is unique. 1 3 2 6 A= 0 1 4 , b = 7 3 5 9 9 Solution: We are trying to determine existence and uniqueness of a solution to
Solutions to problems from section 1.7 0 3 0 2. Determine if the vectors 0 , 5 , 4 are linearly independent. 2 8 1
Solution: Consider the matrix A whose columns are the vectors listed above. The vectors above are linearly independent if and only if the eq
Solutions to even problems from section 4.7 vector( 2. Let B = cfw_b1 , b2 and C = cfw_c1 , c2 be bases for a (hhhhh V , and suppose b1 = c1 +4c2 ( space and b2 = 5c1 3c2 . a. Find the change-of-coordinates matrix from B to C . b. Find [x]C for x = 5b1
Solutions to problems from section 5.1 1 367 6. Is 2 an eigenvector of 3 3 7 ? If so, nd the eigenvalue. 565 1 1 2 367 1 3 3 7 2 = 4 = 2 2 1 2 565 1 1 367 so 2 is an eigenvector of 3 3 7 with eigenvalue 2. 565 1 1 22 8. Is = 3 and eigenvalue of A = 3 2 1
Solutions to problems from section 5.2 2. Find the characteristic polynomial and the eigenvalues of the matrix A = Solution: A () = det 5 3 3 5 = (5 )2 9 = 2 10 + 16 = ( 8)( 2). 53 . 35
Therefore the eigenvalues of A are 8 and 2 (both with multiplicity on
Solutions to problems from section 5.3 2. Let P = 2 3 ,D= 3 5 1 0 , and A = P DP 1 . Find A4 . 0 1/2 53 32 so that 53 32 151/16 45/8 225/16 67/8
Solution: First, since det P = 1 we know that P 1 = A4 = (P DP 1 )4 = P D 4 P 1 = 2 3 3 5 1 0 0 1/16 53 32 2 3
Solutions to problems from section 5.3
subspaces hhhh of R( 2. Let D = cfw_d1 , d2 and B = cfw_b1 , b2 be bases for ( vector spaces hh h
m
V and W , respectively. Let
T : V W be a linear transformation with the property that T (d1 ) = 2b1 3b2 , T (d2 )
Solutions to problems from section 1.3 4. Let u = 3 and v = 2 arrows on an xy -graph: u, Solution: v, v , 2 . Display the following vectors using 1 u + v, u v, and u 2v.
2v,
6. Write a linear system which is equivalent to x1 Solution: 2x1 + 8x2 + x3 = 0,
Solutions to problems from section 1.5 6. Write the solution set of the given homogeneous system in parametric vector form. x1 + 3x2 5x3 = 0 x1 + 4x2 8x3 = 0 3x1 7x2 + 9x3 = 0 Solution: First we nd the general solution: 1 3 5 0 1 3 5 0 1 3 5 0 N R2 =R2 R1
Solutions to even numbered problems from section 1.6 4a. Suppose an economy has four sectors, Agriculture (A), Energy (E), Manufacturing (M), And Transportation (T). Sector A sells 10% of its output to E and 25% to M and retains the rest. Sector E sells 3
Solutions to even problems from section 3.3 2. Use Cramers rule to compute the solutions of the linear system 4x1 + x2 = 6 5x1 + 2x2 = 7 41 . Then det A = 4(2) 5(1) = 3 = 0 so A is invertible. Since A 52 6 ( for i = 1, 2. is invertible we can use Cramers