MATH0 Linear Algebra B Homework 6 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra its Applications Pearson, 006 (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 Hbook 0 (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 reading your lecture notes the recommended text books In respect of MATH0 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 self-control of understing of lecture material Be prepared to answer the following oral questions if asked in the supervision class: (3,, 8, ) In questions 0, the matrix A is n n True of False: If the equation Ax 0 has only the trivial solution, then A is row equivalent to the n n identity matrix If the columns of A span R n, then the columns are linearly independent 3 The equation Ax b has at least one solution for each b R n 3 If the equation Ax 0 has nontrivial solutions then A has fewer than n pivot positions If A T is not invertible then A is not invertible 6 If there is an n n matrix D such that AD I then there is also an n n matrix C such that CA I 6 7 If the columns of A are linearly independent then they span R n 7 8 If the equation Ax b has at least one solution for each b R n, then the solution is unique for each b 8 9 If the linear transformation x Ax maps R n onto R n, then A has n pivot positions 9 f there is b R n such that the equation Ax b is inconsistent, then the transformation x Ax is not one-to-one 0 (8, ) True or False? [Numeration of questions continues from the previous exercise] is the same as the column space of the matrix [ v v p ] If then v,, v p R n Span{ v,, v p } The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of R m
MATH0 Linear Algebra B Homework 6 3 The columns of an invertible n n matrix form a basis of R n 3 Row operations do not affect linear dependence relations among the columns of a matrix The null space of an m n matrix is a subspace of R n 6 The column space of a matrix A is the set of solutions of Ax 0 6 7 If B is a echelon form of a matrix A, then the pivot columns of B form a basis of Col A 7 Submit for marking: 3 (*) Invert the matrices using row operations on the augmented matrix 0 0 0 0 0 0 A 0 0 0 0 0 0, 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 (*) Determine whether these two matrices are invertible Use as few calculations as possible 9 3 A 6 0 3 7, 8 6 0 0 0 0 0 0 0 0 0 0 Invert those of the two matrices which are invertible (*) True or false? If A, A, B B are n n matrices, A A [ B B ], then the product BA is defined, but AB is not Solve the following exercises but do not submit them for marking 6 (33,, 7) Determine which of the following matrices are invertible Use as few calculations as possible Justify your answers 0 0 0 7 3 8 0 3 0, 9 7 3 0 3 8 3 6 3 0 8 (3, 6, 7) The following questions can be instantly answered with the help of the Invertible Matrix Theorem Can a square matrix with two identical columns be invertible? Why or why not? Is it possible for matrix to be invertible when its columns do not span R? Why or why not? 3 If A is invertible then the columns of A are linearly independent Explain why If the columns of a 6 6 matrix D are linearly independent, what can you say about solutions of Dx b? Why? 7 (33) When is a square upper triangular matrix invertible? Justify you answer 9 (333) In this exercise, T is a linear transformation from R to R Show that T is invert-
MATH0 Linear Algebra B Homework 6 3 ible find a formula for T () x x + 9x T x x 7x 0 (, 7) Find formulae for matrices X, Y Z in terms of square matrices A, B, C, given the following equalities between partitioned matrices: A B I 0 C 0 X Y [ X 0 0 Y ] Z A 0 0 B I ; Z 0 I 0 Answers to True/False questions True True 3 False True True 6 True 7 True 8 True 9 True 0 True True False; it is a subspace of R n 3 True True True 6 False 7 False It is pivot columns of A (that is, columns of A positioned the same way as pivot columns of B) that form a basis of Col A Solutions for non-starred exercises 6 (33,, 7) Determine which of the following matrices are invertible Use as few calculations as possible Justify your answers 0 0 0 7 3 8 Solution: Invertible Swap rows 3: the matrix will become upper triangular will obviously have a pivot position in each column 0 3 0, 9 7 Solution: Not invertible After row operations R 3 R 3 + R R 3 R 3 + 3R the bottom row is filled with zeroes only 3 0 3 8 3 6 3 0 Solution: Invertible The matrix reduces to an echelon form with pivot positions 7 (33) When is a square upper triangular matrix invertible? Justify you answer Solution: When its diagonal elements are all non-zero 8 (3, 6, 7) The following questions can be instantly answered with the help of the Invertible Matrix Theorem Can a square matrix with two identical columns be invertible? Why or why not? Solution: No, because its columns are linearly dependent Is it possible for matrix to be invertible when its columns do not span R? Why or why not? Solution: No 3 If A is invertible then the columns of A are linearly independent Explain why Solution: Because A is also invertible If the columns of a 6 6 matrix D are linearly independent, what can you say about solutions of Dx b? Why? Solution: It exists unique for every b R 6
MATH0 Linear Algebra B Homework 6 9 (333) In this exercise, T is a linear transformation from R to R Show that T is invertible find a formula for T T ([ x x ]) x + 9x x 7x Solution: The stard matrix for T is A Since det A, hence A 9 7 7 9 7 9, () T x 7x + 9x x x + x 0 (, 7) Find formulae for matrices X, Y Z in terms of square matrices A, B, C, given the following equalities between partitioned matrices: A B I 0 C 0 X Y Solution: We get four equations AI + BX 0 A 0 + BY I CI + 0 X Z C 0 + 0 Y 0, ; Z 0 the last of which is equivalent to 0 0 can be deleted, while the rest can be simplified as A + BX 0 BY I C Z Notice that BY I means that B is invertible Y B But then X can also be found as X B A Z X 0 0 A 0 0 Y B I Solution: We get the system XA I XZ 0 Y A + 0 Y Z + I I, I 0 from which we conclude that X A are invertible X A, since X is invertible, Since A is invertible, Z X 0 0 Y BA
MATH0 Linear Algebra B Homework 6 Solutions for starred exercises 3 (*) Invert the matrices using row operations on the augmented matrix 0 0 0 0 0 0 A 0 0 0 0 0 0, 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 Answer: 0 A 0 0 0 0 0, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 (*) Determine whether these two matrices are invertible Use as few calculations as possible 9 3 A 6 0 3 7, 8 6 0 0 0 0 0 0 0 0 0 0 Invert those of the two matrices which are invertible Answer: A is not invertible since columns are linearly dependent: for example, a a a 3 a therefore a a (a 3 a ) a + a a 3 0 The matrix B is invertible: a permutation of rows according the rule smaller the number of s in the row higher it foes killing s below the pivot entries obviously turns it into the identity matrix 0 0 0 B 0 0 0 0 0 0 0 0 0 (*) True or false? If A, A, B B are n n matrices, A B A B, then the product BA is defined, but AB is not Answer: False, since both products AB BA are defined: BA A B B B A A + B A, while A [ A ] [B ] B B A B A B A B