MATH Linear Algebra B Homework Week 5 Students are strongly advised to acquire a copy of the Textbook: D C Lay Linear Algebra its Applications Pearson 6 (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 (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 MATH 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 class: (9 4 5 6 9 ) True of False: A linear transformation T : R n R m is completely determined by its effect on the columns of the n n identity matrix When two linear transformations are performed one after another the combined effect may not always be a linear transformation A mapping T : R n R m is onto R m if every vector x in R n is mapped onto some vector in R m 4 If A is a matrix then the transformation x Ax cannot be one-to one 4 5 Not every linear transformation from R n to R n is a matrix transformation (that is has the form x Ax for some matrix A) 5 6 matrix then the transformation x Ax cannot map R onto R 6 7 If A B are with columns a a b b respectively then 7 AB = [ a b a b 8 Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A 8 9 AB + AC = A(B + C) 9
MATH Linear Algebra B Homework Week 5 A T + B T = (A + B) T A T B T = (AB) T If A B are then B = [ b b b AB = [ Ab + Ab + Ab The second row of AB is the second row of A multiplied on the right by B 4 In order for a matrix B to be the inverse of A both equations must be true 4 AB = I B I 5 If A B are n n invertible then A B is the inverse of AB 5 6 If [ a b c d ab cd then A is invertible 6 7 If A is an invertible n n matrix then the equation Ax = b is consistent for each b R n 7 8 Each elementary matrix is invertible 8 9 If A is invertible then elementary row operations that reduce A to the identity I n also reduce A to I n 9 Answers to True/False questions True False False 4 False A counterexample is indeed the map [ x is one-to-one 5 False ; [ x = x x x x 6 True; the system Ax = b with a coefficients matrix A has at most two pivot positions therefore cannot have a solution for each b R 7 False 8 False In the correct formulation the roles of A B have to be changed: each column of AB is a linear combination of the columns of A using weights from the corresponding column of B 9 True True False False True 4 True 5 In general False 6 False One of many counterexamples: (Why?) 7 True 8 True [ 9 False Actually the row operations that reduce A to I reduce A to A A = A (why?) which is not generally equal I
MATH Linear Algebra B Homework Week 5 Solve but do not submit for marking: (8) Let define by [ T : R R T (x) = Ax Find the images under T of [ u = v = [ a b (8 5) With T defined by T (x) = Ax find a vector x whose image under T is b determine whether x is unique (a) 6 b = 7 ; 5 (b) [ 5 7 b = 7 5 [ 4 (87) Let A be a 6 5 matrix What must a b be in order to define by T : R a R b T (x) = Ax? 5 (89) Find all x R 4 that are mapped into the zero vector by transformation x Ax for 4 7 5 4 6 6 4 Again this is a system of linear equations in disguise 9 7 x = x 4 + x4 6 (8) Let b = A as in the previous problem Is b in the range of the linear transformation x Ax? Why or why not? 7 (87) Let T : R R be a linear transformation that maps u = [ 5 [ v = [ [ Use the fact that T is linear to find the images under T of u v u + v 8 (89) Let e = let [ e = [ y = T : R R [ y 5 = be a linear transformation that maps Find the images of e y e y [ 5 9 (8) Let [ x x T : R n R m [ 6 be a linear transformation let { v v v } be a linearly dependent set in R n Explain why the set { T (v ) T (v ) T (v ) } is linearly dependent
MATH Linear Algebra B Homework Week 5 4 (Not in the textbook) Explain why the transformation ([ ) x + x x T = x x x is not linear (95 6) Fill in missing entries in the matrices assuming that the equations hold for all values of the variables????????? x x x?? [?? x x?? = = x x 4x x x + x x x x + x x (97 9) Show that T is a linear transformation by finding a matrix which implements the mapping x T x x = x 4 x T x = x (9) Let x + x x + x x + x 4 [ x 5x + 4x T : R R x 6x be a linear transformation such that ([ ) [ x x + x T = x 4x + 5x Find x such that T (x) = 4 (9 5) Let [ 8 T : R n R m be a linear transformation A its stard matrix (a) Complete the following statement to make it true: T is one-to-one if only if A has pivot columns Explain why this statement is true (b) If T maps R n onto R m can you give a relation between m n? (c) If T is one-to-one what can you say about m n? 5 () Let [ 4 5 Compute I A (I )A 6 (5) If a matrix A is 5 the product AB is 5 7 what is the size of B? 7 (9) Let [ 5 B = [ 4 5 k What value(s) of k if any will make AB = BA? 8 () Let D = 4 5 5 Compute AD DA Explain how the columns or rows of A change when A is multiplied by D on the right or on the left Find a matrix B not the identity matrix or the zero matrix such that AB = BA 9 (7) If [ 5 AB = [ 6 9 determine the first second column of B
MATH Linear Algebra B Homework Week 5 5 (9) Suppose the third column of B is the sum of the first two columns What can you say about the third column of AB? Why? () Suppose C I n (the n n identity matrix) Show that the equation Ax = has only the trivial solution Explain why A cannot have more columns than rows (5) Suppose A is an m n matrix there exist n m matrices C D such that C I n AD = I m Prove that C = D [Hint: think of CAD (7) Let a u = v = b 4 c Compute u T v v T u uv T vu T 4 ( 5) Find the inverse of [ 8 6 5 4 use it to solve the system 5 (7) Let b = 8x + 6x = 5x + 4x = [ b = [ 5 [ b 5 = [ b 6 4 = [ 5 (a) Find A use it to solve Ax = b Ax = b Ax = b Ax = b 4 (b) The four equations in (a) can be solved by the same set of row operations since the coefficient matrix is the same in each case Solve the four equations in part (a) by row reducing the augmented matrix [ A b b b b 4 6 (9) If A B C are n n invertible matrices does the equation C (A + X)B = I n has a solution X? If so find it 7 (9 ) Find the inverses of this matrices if they exist: [ 4 7 4 4 8 () Find the inverses of Let A be the corresponding n n matrix let B be its inverse Guess the form of B then prove that AB = I B I 9 (5) Let 7 9 5 6 4 Find the third column of A without computing the other columns
MATH Linear Algebra B Homework Week 5 6 Solutions to non-starred exercises (8) 7 (87) T (u) = [ 6 T (v) = [ a b [ [ [ 6 = = [ 6 [ [ + = [ 4 9 (8 5) 8 (89) This amounts to solving a system of equations with the coefficient matrix A the right part b Answers are (a) Unique solution x = (b) Solution is not unique one of many solutions is x = 4 (87) [ 7 9 (8) If [ x x 5x + 6x c v + c v + c v = with at least one of weights c c c not equal zero then c T (v ) + c T (v ) + c T (v ) = T () = therefore the set { T (v ) T (v ) T (v ) } a = 5 b = 6 is linearly dependent 5 (89) Again this is a system of linear equations in disguise 9 7 x = x 4 + x4 6 (8) Yes because the system of linear equations Ax = b is consistent: check this by transforming the augmented matrix [ 4 7 5 A b = 4 6 6 4 into echelon form (Not in the textbook) Compute for example ( [ ) 6 T = 4 T (95 6) 4 (97 9) ([ ) 6 = 4
MATH Linear Algebra B Homework Week 5 7 [ 5 4 6 (9) 4 (9 5) x = [ 7 4 9 (7) (9) The same The first three columns of AB are Ab Ab Ab since the first three columns b b b of B satisfy b + b = b we also have Ab + Ab = Ab (a) n; (b) m n; (c) n m () 5 () [ 5 5 6 (5) [ 5 6 Multiply the both sides of Ax = by C on the left: CAx = C I nx = which means x = The matrix A cannot have more columns than rows for otherwise the equation 7 7 (9) Ax = would have free unknowns infinitely many solutions k = 5 For solution observe that the matrix equation AB = BA amounts to a system of 4 linear equations one equation for each entry in AB in a single unknown k Check that this system is consistent has a unique solution k = 5 8 () 5 AD = 6 5 D 6 9 5 5 5 When A is multiply by D on the right the columns of A are multiplied by the corresponding diagonal values of D; when A is multiply by D on the left the rows of A are multiplied by the corresponding diagonal values of D For B you can take for example B = A = AA Or B = A Or even B = A n for any integer n > (5) We can rearrange the product CAD in two ways: CAD = C(AD) = CI m = C Hence C = D (7) CAD = (CA)D = I nd = D u T v = a + b 4c v T u = a + b 4c a b c uv T = a b c 4a 4b 4c a a 4a vu T = b b 4b c c 4c
MATH Linear Algebra B Homework Week 5 8 4 ( 5) [ x x 5 (7) [ [ 8 6 = 5 4 5 4 [ [ [ 9 = 5 = 4 9 x = 9 x = 9 The solutions of the four equations are [ [ [ [ 9 6 4 5 5 while 6 (9) A = [ 5 Multiplying the both parts of equation by C on the left B on the right one gets A + X = CI nb X = CB A After that we have to check by substitution that X is a solution 7 (9 ) 8 () [ 7 4 8 4 B = 7 This pattern becomes obvious after finding the inverses in 4 4 cases Checking AB = I B I is straightforward 9 (5) Work with the augmented matrix [ 7 9 A e = 5 6 4 use elementary row operations to produce the identity matrix I in place of the first three columns Answer: 6 4 PTO
MATH Linear Algebra B Homework Week 5 9 Submit for marking: (*) Let [ 4 6 Construct a matrix B such that AB is the zero matrix Use two different nonzero columns for B (*) Suppose that the first two columns b b of B are equal What can be said about the columns of AB (if AB is defined)? Why? (*) Suppose P is invertible P BP Solve for B in terms of A (*) Repeat the strategy of the Exercise 8 to guess the inverse of n Prove that your guess is correct Solutions for starred exercises: (*) Let [ 4 6 Construct a matrix B such that AB is the zero matrix Use two different nonzero columns for B (*) Suppose P is invertible P BP Solve for B in terms of A Answer: B = P AP Answer: For example [ 4 B = Observe that in all possible examples columns of B are scalar factors of [ (*) Find the inverse of Answer: A = 6 6 (*) Suppose that the first two columns b b of B are equal What can be said about the columns of AB (if AB is defined)? Why? Answer: The first two columns of AB are equal because they are both equal Ab = Ab 4 (*) Repeat the strategy of the Exercise 8 to guess the inverse of n Prove that your guess is correct
MATH Linear Algebra B Homework Week 5 Answer: After computing A for n = 4 the pattern becomes obvious: A = n n Checking that AA = I is straightforward can be done by mental arithmetic if you know where to look What follows is a detailed write-up Denote = B n n Denote the rows of the matrix A by A A n columns of B by B B n to that A A n B = [ B B n If now C = AB then its element c ij in row i column j equals c ij = A i B j a product of n (row) matrix A i n (column) matrix B j If i < j then c ij is positioned above the diagonal in C; i = j then c ij is positioned on the diagonal in C; i = j then c ij is positioned below the diagonal in C; In the first case c ij = [ i j j+ every non-zero element in A i is matched in the product with a zero in B j Therfeore c ij = if i < j I leave you to check on your own that in a bit trickier way that After that of course c ij = if i > j c ii = C = I There is a much simpler solution based on direct reduction of Problem 4 to Problem 8 Rename the matrix given in Problem 8 as A [from 8 but retain A as the notation for the matrix from Problem 4 Observe that if D = n is the diagonal n n matrix with numbers n on the diagonal then therefore A [from 8 D A = D A [from 8 = n n