MATH10212 Linear Algebra B Homework 7


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1 MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments will consist of some odd numbered exercises from the Textbook The Textbook contains answers to most odd numbered exercises The Student Handbook 2 (f) says: As a rough guide you should be spending approximately twice the number of instruction hours in private study, mainly working through the examples sheets and reading your lecture notes and the recommended text books In respect of MATH22 Linear Algebra B this means that students are expected to spend 8 (eight!) hours a week in private study of Linear Algebra The homework is set as an approximately two hours task of written work, plus oral questions where workload is harder to quantify these questions serve mostly for selfcontrol of understanding of lecture material Be prepared to answer the following oral questions if asked in the supervision class: (Mostly 3227, 28) Answer True or False The determinant of a zero matrix is zero 2 A row replacement operation does not affect the determinant of a matrix 2 3 If the columns of A are linearly dependent, then 3 det A = 4 det(a + B) = det A + det B 4 5 If two row interchanges are made in succession then the new determinant equals the old determinant 5 6 The determinant of A is the product of diagonal entries of A 6 7 If det A = then two rows or two columns are the same, or a row or a column is zero 7 8 det A T = ( ) det A 8 9 If A is an n n matrix then 9 det(a + A) = 2 n det A
2 MATH22 Linear Algebra B Homework 7 2 If A is an n n matrix then det A T = ( ) n det A Answers to True/False questions True 2 True 3 True 4 False 5 True 6 False 7 False 8 False 9 True False Submit for marking: 2 (*) What can you say about the shape of an m n matrix A when the columns of A form a basis of R m? 3 (*) Find the determinant by row reduction to echelon form det (*) Combine the methods of row reduction and cofactor expansion to compute the determinant 2 3 det (*) Let A and P be n n matrices, with P invertible Show that det(p AP ) = det A Solve the following exercises but do not submit them for marking 6 (285) Let v = 3, v 2 = 5, and w = Determine if w is in subspace if R 3 generated by v and v 2 7 (287, 9) Let v = 8, v 2 = 8, v 3 = 6, p =, and A = [ v v 2 v 3 ] (a) How many vectors are in { v, v 2, v 3 }? (b) How many vectors are in Col A? (c) Is p in Col A? Why or why not? (d) Determine if p is in Nul A 8 (28, 3) Give integers p and q such that Nul A is a subspace of R p and
3 MATH22 Linear Algebra B Homework 7 3 Col A is a subspace of R q : A = Find a nonzero vector in Nul A and a nonzero vector in Col A 9 (285, 7, 9) Determine which sets are bases for R 2 or R 3 : {[ ] [ ]} 5, ; , 7, 3 ; , 2 5 (2823) Shown are a matrix A and an echelon form for A: A = Find a basis for Col A and a basis for Nul A (2827, 28, 29) Construct a 3 3 matrix A and a nonzero vector b such that (a) b is in Col A, but b is not the same as any one of the columns of A; (b) b is not in Col A; (c) b is in Nul A 2 (283 36) Respond as comprehensively as possible, and justify your answer (a) Suppose F is a 5 5 matrix whose column space is not equal to R 5 What can you say about Nul F? (b) If R is a 6 6 matrix and Nul R is not the zero subspace, what can you say about Col R? (c) If Q is a 4 4 matrix and Col Q = R 4, what can you say about solutions of equations of the form for b R 4? Qx = b (d) If P is a 5 5 matrix and Nul P is the zero subspace, what can you say about solutions of equations of the form P x = b for b R 5? (e) What can you say about Nul B when B is a 5 4 matrix with linearly independent columns? 3 (29) Find the vector x determined [ ] by the given coordinate vector x and the given basis B: B {[ ] [ ]} [ ] 2 [ ] 3 B =,, x = B 2 4 (293, 5) The vector x is in subspace H with a basis B = {b, b 2 } Find the Bcoordinate vector of x [ ] [ ] [ ] 2 3 b =, b 4 2 =, x = ; b = 5, b 2 = 7, x = The following exercises use the following notation for determinants: a b [ ] a b c d = det c d a b c a b c d e f = det d e f, g h i g h i etc
4 MATH22 Linear Algebra B Homework (3, 3, 5) Compute the determinants using a cofactor expansion (a) across the first row, and (b) across the second column ; ; Solution: ; 5; 23 6 (39,, 3) Compute the determinants by cofactor expansion At each step, choose a row or column that involves the least amount of computation ; ; (329) Compute the determinant by row reduction to echelon form: (323) Combine the methods of row reduction and cofactor expansion to compute the determinant: (Mostly ) Given that a b c d e f g h i = 7, find the following determinants: (a) (b) (c) (d) (e) (f) (g) a b c d e f 5g 5h 5i a c b d f e g i h a b c g h i d e f a d g b e h c f i a b c 2d + a 2e + b 2f + c g h i a b c d e f g h i 2a 2b 2c 2d 2e 2f 2g 2h 2i 2 (3232) Find a formula for det(ra) where A is a n n matrix and r is a scalar 2 (3239) Let A and B be 3 3 matrices, with det A = 4 and det B = 3 Use properties of determinants to compute: (a) det AB;
5 MATH22 Linear Algebra B Homework 7 5 (b) det 5A; (c) det B T ; (d) det A ; (e) det A 3 22 (333, 5) Use Cramer s rule to compute the solutions of the following systems: (a) (b) 3x 2x 2 = 7 5x + 6x 2 = 5 2x + x 2 = 7 3x + x 3 = 8 x 2 + 2x 3 = 3 23 (337) Use Cramer s rule to determine the values of the parameter s for which the system has a unique solution, and describe the solution: 6sx + 4x 2 = 5 9x + 2sx 2 = 2 24 (33) Compute the adjugate of the given matrix, and then use it to give the inverse of the matrix: 2 3 Solutions to nonstarred exercises 6 Solution: No The system is inconsistent x v + x 2 v 2 = w 7 Solution: (a) Three vectors: v, v 2, and v 3 (b) Infinitely many vectors (c) Yes, because Ax = p has a solution (d) No, because Ap 8 Solution: p = 4 and q = 3 For a nonzero vector in Nul A chose, for example, [ 2 ] T or any other nontrivial solution x of the homogeneous system Ax = For a nonzero vector in Col A select, for example, any column of A 9 Solution: The first two sets The third set cannot be a basis of R 3 because it contains only two vectors, while each basis of R 3 contains three linearly independent vectors Solution: For a basis of Col A, you can take the two pivoted columns of A: 4 5 6, Indeed, nonpivoted columns are linear combinations of pivoted columns (see the solution to Problem 26(b) in Homework 4) On the other hand, it is obvious that pivoted columns are linearly independent Therefore pivoted columns form a basis of Col A (it was discussed in the lectures) One of possible bases for Nul A is 4 7 5, 6 Indeed, to find a basis for Nul A is the same as to solve the homogeneous system of equations AX = Let us complete reduction of A to its reduced echelon form:
6 MATH22 Linear Algebra B Homework 7 6 Variables x 3 and x 4 are free, denote them x 3 = s and x 4 = t while x = 4x 3 5x 4 = 4s 5t x 2 = 5x 3 + 6x 4 = 5s + 6t We see now that an arbitrary solution is uniquely written as a linear combination x 4s 5t 4 7 x 2 x 3 = 5s + 6t s = s 5 + t 6 x 5 t which means that 4 5, 7 6 is a basis of the solution space of the solution space of Ax =, that is, of Nul A Solution: The problem has infinitely many possible answers One of them is A =, then the vector belongs to Col A but is not equal to any of the columns of A, while the vector b = simultaneously does not belong to Col A (because the system Ax = b is inconsistent) and belongs to Nul A (because Ab = 2 Solution: (a) Nul F is not the zero subspace (b) Col R R 6 (c) Solutions exist and are unique for each b R 4 (d) Solutions exist and are unique for each b R 5 (e) Nul B is the zero subspace of R 5 3 Solution: x = 3v + 2v 2 = [ ] 7 4 Solution: [ ] 7, 5 [ ] /4 5/4 5 Solution: ; 2; 6 7 Solution: 3 8 Solution: 6 9 Solution: (a) 35; (b) 7; (c) 7; (d) 7; (e) 4; (f) 7; (g) 56 2 Solution: det(ra) = r n det A 2 Solution: (a) 2; (b) 5; (c) 3; (d) /4; (e) Solution: (a) (b) [ ] 4 ; 5/2 3/2 4 7/2 23 Solution: s ± 3; x = 5s + 4 6(s 2 3) ; x 4s 5 2 = 4(s 2 3) 24 Solution: adj A = 3 3, A =
7 MATH22 Linear Algebra B Homework 7 7 Solutions for starred exercises 2* Answer: n = m 3* Answer: ; in the process of row reduction a zero row will pop up fairly soon, because of linear dependency of rows: R 3 = 2R R 2 4* Answer: 4 5* Answer: det(p AP ) = det P det A det P = det A
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