5 and A,1 = B = is obtained by interchanging the rst two rows of A. Write down the inverse of B.

Transcription

1 EE { QUESTION LIST EE KUMAR Spring (we will use the abbreviation QL to refer to problems on this list the list includes questions from prior midterm and nal exams) VECTORS AND MATRICES. Pages - of the text, Problem..8. Pages - of the text, Problem... Pages - of the text, Problem... Pages - of the text, Problem... Pages - of the text, Problem... Pages - of the text, Problem... Pages - of the text, Problem.. 8. The matrix A as well as it's inverse A, are given below: The matrix,,, B = and A, =,,,,,,,,,,,, is obtained by interchanging the rst two rows of A. Write down the inverse of B. Hint: Write B = PA for a suitable matrix P. 9. Prove that (AB) T = B T A T.. Exercise.. of the text, page. :

2 GAUSIAN ELIMINATION AND LDU FACTORIZATION. Express the matrix in the form P LDW where,, 9 P isa( ) permutation matrix L isa( ) lower triangular matrix with 's along the diagonal D is a diagonal matrix W is a ( ) upper triangular matrix with 's along the diagonal, i.e., nd the matrices P, L, D and W.. The equation below describes the row-reduction of an n n matrix A using elementary row operations:, {z } E, {z } E {z } E Write downa( ) matrix L such that LU.. Exercise.., page of the text., {z } A =, {z } U Cholesky decomposition, triangular matrices, singularity, inverse Computation :. Is the matrix singular?,,,,,. Use the Gauss-Jordan method to nd the inverse of the matrix, 9

3 . \A typical Chinese problem, taken from the Han dynasty text Nine Chapters of the Mathematical Art (about B.C.) reads: There are three classes of corn, of which three bundles of the rst class, two of the second, and one of the third, make 9 measures. Two of the rst, three of the second and one of the third, make measures. And one of the rst, two of ther second and three of the third make measures. How many measures of grain are contained in one bundle of each class? The system of equations here is: x +y + z = 9 x +y + z = x +y +z = :" {Taken from notes of Victor Katz appearing in Linear Algebra by J.B.Fraleigh and R. A. Beauregard. Solve the problem using the augmented matrix formulation.. Factor the symmetric matrix " 8 # : into a product SS T matrix. (Cholesky decomposition), where S isalower triangular 8. Exercise.., page 9 of the text. Vector Spaces, subspaces 9. Let V beavector space. Show that if v is in V and r is a scalar and if rv =, then either r = or else, v =.. Identify geometrically, as clearly as you can, the subset of -dimensional (Euclidean) space < that corresponds to the column space of the matrix. Let S be the subset of the (x,y)-plane (i.e., the subset of < ) given by : Is S a subspace of the (x,y)-plane? S = f(x; y) j x +y =g:

4 . Consider the vector space V = < with addition dened by and scalar multiplication dened by [x; y] [w; z] = [x + w +;y+ z, ] a [x; y] = [ax + a, ;ay, a +]: Does V with these operations, form a vector space? If your answer is yes, identify the identity element and the additive inverse of each element. If your answer is no, explain clearly, whyyou feel this fails to be a vector space. Start by stating all the axioms that dene a vector space and your conclusions on each of the axioms. (CARE { Do not jump to conclusions on this problem).. Which of the following are examples of subspaces? (In each case, the operations of vector addition and scalar multiplication are inherited from the parent vector space). (a) the set fc + w j w W g; where c is a xed nonzero element of< n and W is a xed subspace of < n, (b) the set of all ( ) nonsingular matrices (c) the set of all polynomials of the form where c ;c ;c are real numbers, c + c X + c X + X (d) the set of all odd functions f : <!<, i.e., functions f(x) such that (e) none of the above. Let H be the matrix H = f(,x) =,f(x) 8x <: Assume that you are working in the eld F of two elements f; g. Addition, denoted by, operates as follows = = = =

5 Multiplication of elements in the eld, denoted by, operates in the usual way, i.e., = = = = Determine all the vectors in the nullspace of the matrix H. (Hint: the nullspace is in this case a nite set!) Columnspace, nullspace, rowspace and SLE. Which of the following are examples of subspaces? (In each case, the operations of vector addition and scalar multiplication are inherited from the parent vector space). (a) the set fc + w j w W g; where c is a xed nonzero element of< n and W is a xed subspace of < n, (b) the set of all ( ) nonsingular matrices (c) the set of all polynomials of the form where c ;c ;c are real numbers, c + c X + c X + X (d) the set of all odd functions f : <!<, i.e., functions f(x) such that (e) none of the above f(,x) =,f(x) 8x <:. Identify geometrically, as clearly as you can, the subspaces of -dimensional (Euclidean) space < that correspond (i) to the rowspace and (ii) to the nullspace of the matrix:,,, Do you notice any relationship between the two?. Exercise.., page of the text. (You may ignore the part of the question that is within parenthesis). 8. For the matrix given in Review Exercise., page 8 of the text, determine the echelon form. (You are not required to nd the dimension of its four fundamental subspaces). :

6 9. How many possible patterns can you nd (like the one in Fig.., page of the text) for ( ) echelon matrices. Entries to the right of the pivots are irrelevant.. Find necessary and sucient conditions on the components b ;b ;b and b under which the system of linear equations will have at least one solution.,,, x x x x = SLE { the general case, linear independence. Find conditions on b ;b ;b such that the system of linear equations x +y + z +w = b x + y +z +8w = b x + y +:z +w = b will have at least one solution. Given that b =,;b =,choose b such that the system of linear equations has at least one solution (it turns out that this is possible in this case). Then for this choice of b ;b ;b, nd all solutions to the system of linear equations. Follow the steps outlined in class: (i) rst identify the augmented matrix; then reduce the left-hand side of this augmented matrix to echelon form; nd the condition(s) for existence of a solution. Then choose b appropriately, identify a particular solution and the nullspace of the coecient matrix and nally, write down an expression for the set of all solutions.. Row-reduce to echelon form the coecient matrix of the system of linear equations: x, x + x +x =; x + x, x + x =; x +x, x +x =: and hence nd the most general solution. (Follow the steps outlined in Problem above).. Are the vectors v = ;v = ;v = b b b b and v = in < linearly independent? Show all intermediate steps and explain your answer.

7 . If the vectors u, v and w of a vector space V are linearly independent, does it necessarily follow that the vectors u + v, v + w and w + u will always be linearly independent?. For the system of linear equations presented in Exercise..8, page 8 of the text, after nding an equation that c must satisfy in order for there to be a solution, use this equation to identify geometrically, the column space of the coecient matrix A.. Exercise.., page 9 of the text. (Hint: rst determine the coecient matrix based on the given nullspace and the condition b + b = b for existence of a solution. Then nd a right-hand side for which [ ] T is a particular solution.) Spanning, Basis, coordinate representation, change of basis. Exercise.., page 8 of the text. 8. Find a basis each for the rowspace, columnspace and nullspace of the matrix, : 9. Find a basis for the vector space consisting of all polynomials f(x) =a + a x + a x of degree less than or equal to that satisfy in addition, the condition that their coecients sum to zero, i.e., a + a + a =.. What is the coordinate representation of the vector [ ] T with respect to the basis n [ ] T ; [ ] T ; [ ] To?. Consider the following two basis for < : (" # ; " #) and Find a matrix P such that for all x <. B = [x] B (" # ; " = P [x] A #)

8 . Consider the following two basis for the vector space of all ( ) upper triangular matrices U: (" # " # " #) ; ; and B = Find a matrix P such that (" for all upper triangular matrices U. Dimension, Rank # ; [U] B ", = P [U] A # ; ". The nullspace of a certain ( ) matrix is a plane in < that passes through the origin. What is the rank of A?. The ( ) matrix A has rank. Which of the following is true of A? (a) If a solution to Ax = b exists, then that solution is unique. (b) Ax = b always has at least one solution (c) Neither of the above two statements is necessarily true. Let A be an (m n) matrix and consider the system of linear equations Ax = b. Let B =[A j b] be the (m n + ) augmented matrix. Then Ax = b has at least one solution if and only if the matrices A and B have the same rank. #) (i) True (ii) False. What is the dimension of the vector space of all upper triangular matrices? (A is upper triangular if and only if a = a = a = ).. Exercise.8, page 9 of the text. 8. Exercise., page 9 of the text. 9. Reduce the matrix A below toechelon form. Use the echelon form to indicate with a ( p ) all the statements below that are true concerning the matrix A. (a) A has a left inverse,,,,,, 8 :

9 (b) A has a right inverse (c) For every b <, Ax = b has a solution. (d) If Ax = b has a solution, then that solution is unique. (e) none of the above is true of A.. Exercise.., page of the text. Explain your answers. Geometric Notions, Least-squares Approximation. Let A be an (m n) matrix and b be an (m ) vector. (a) Under what conditions on A does the equation A t Ax = A t b have a solution x? (b) Under what conditions on A does the equation A t Ax = A t b have a unique solution x (given that a solution exists)?. Let P denote the plane in < containing the vectors x =[] t and y =[] t. Find avector z in P that is orthogonal to x.. The result of projecting the vector z =[] t onto the plane Q in < is the vector z =[] t. Identify as clearly and as explicitly as you can, the plane Q.. Find the quadratic polynomial y = ax + bx + c that is the best least-squares t to the points (,;,); (,; ); (; ); (; ); (;,) :. Exercise..8, page of the text.. Exercise.., page of the text.. Exercise.., page of the text. Spanning, Basis, coordinate representation, change of basis 8. Exercise.., page 8 of the text. 9. Find a basis each for the rowspace, columnspace and nullspace of the matrix, : 9

10 . Find a basis for the vector space consisting of all polynomials f(x) =a + a x + a x of degree less than or equal to that satisfy in addition, the condition that their coecients sum to zero, i.e., a + a + a =.. What is the coordinate representation of the vector [ ] T with respect to the basis n [ ] T ; [ ] T ; [ ] To?. Consider the following two basis for < : (" # ; " #) and Find a matrix P such that for all x <. B = [x] B (" # ; " = P [x] A #). Consider the following two basis for the vector space of all ( ) upper triangular matrices U: (" # " # " #) ; ; and B = Find a matrix P such that (" for all upper triangular matrices U. # ; [U] B ", = P [U] A # ; " LSA, orthogonality, Gram-Schmidt process, complex vectors. Find the least-squares approximate solution of the overdetermined system:, x =, : #)

11 . Apply the Gram-Schmidt orthogonalization process and derive an orthonormal basis for <, starting with the three vectors. Consider the least-squares approximation problem Ax = b, where ; ;, 8 8 and b =[,; ; ; ; ] T. Use the Gram-Schmidt orthogonalization process to replace the second column vector of the matrix A by avector that is orthogonal to the rst column vector. Let B denote the new matrix. Solve Ax = b by solving instead Bx = b instead. Why is this an easier problem? Why is this procedure justied?. Exercise..9 of the text, page. 8. Let w i ;i=; ; ;, denote the four column vectors of the matrix,,,,,, Note that the four column vectors are orthogonal and therefore, linearly independent. (a) Express the vector x = [; ; ; ] t as a linear combination of the four column vectors. (b) Compute the inner products x t w i ;i=; ; ;. (c) Are the above two results related? Can you generalize these results? COMPLEX VECTORS. EIGENVALUES AND EIGENVECTORS 9. Find the lengths of the vectors a =[; ;{] T, b =[+{;, {; +{] T as well as their inner product. Find a vector [x; y; z] T that is orthogonal to both vectors.. Find the eigenvalues of a ( ) matrix A whose trace equals and whose determinant equals,. : :

12 . The rst two terms of a sequence are a = and a =. Subsequent terms are generated using a k = a k, + a k, ; for k : Find a.. Determine whether or not the matrix,,, : is diagonalizable. If A is diagonalizable, express A in the form EE,, where is a diagonal matrix.

13 PRIOR MIDTERM EXAMS unless otherwise specied, all vectors are column vectors EE KUMAR on some multiple-choice questions, more than one choice may be correct; unless otherwise instructed, indicate with a ( p ) all the correct answers you can spot multiple-choice questions will be graded based only upon your answer clearly identify all your nal answers Sp 998. Let V be the vector space of all functions f from < to <. Which (if any) of the following sets of functions are subspaces of V? the subset of all f such that f(x )=(f(x) ) the subset of all f such that f() = f() the subset of all f such that f(,) =. Consider two subspaces V and W of <. (a) Is the intersection V \ W necessarily a subspace of <? (b) Is the union V [ W necessarily a subspace of <? points points. Consider a linear system of four equations with three unknowns. We are told that the system has a unique solution. What does the reduced-row echelon form of the coecient matrix of this system look like? Use to denote a pivot and to denote an entry that may ormay not be zero.. points (a) Let A be a ( ) matrix and let b and c be two vectors in <.We are told that the system Ax = b is inconsistent, i.e., does not have a solution. What can you say about the number of solutions of the system Ax = c? Be as precise as you can possibly be in your answer.

14 (b) Let A be a ( ) matrix and let b and c be two vectors in <.We are told that the system Ax = b has a unique solution. What can you say about the number of solutions of the system Ax = c? Be as precise as you can possibly be in your answer. points. If the n columns of an n n matrix A are linearly dependent, the same will always be true of the n columns of the matrix A. (i) True (ii) False points 8. Singularity of the n n matrix A implies that the matrix A does not have aninverse for every b in < n, the system of equations Ax = b always has an innite number of solutions. the columns of A are linearly dependent none of the above Identify all the correct implications you can spot. points 9. The equation below describes the row-reduction of a matrix A using elementary row operations: {z } P, {z } E {z } P {z } A =, {z } U Determine the matrices P, D and W in the P LDW decomposition of A. 8. Consider the linear system x + y, z = x +y + z = x + y +(k, )z = k : points

15 where k is an arbitrary constant. For which choice(s) of k does this system have a unique solution? For which choice(s) of k does the system have innitely many solutions? For which choice(s) of k is the system inconsistent? points 8. Identify all cubic polynomials (i.e., polynomials of degree ) of the form f(x) =a x + a x + a x + a for some real coecients a i such that the graph y = f(x) passes thru the two points (; ) and (;,) in the (x; y)-plane. points 8. Let V be the vector space of all functions f : <!<. Let f, f and f be functions satisfying f (x )=; f (x )=; f (x )= f (x )=; f (x )=; f (x )= f (x )=; f (x )=; f (x )=9 Are the functions f ;f ;f linearly independent (i.e., is there a nontrivial linear combination of these functions that will yield the function representing the zero vector in V )? Explain your answer and show all intermediate steps clearly. points 8. Let V and W be subspaces of < spanned by andvectors respectively, as shown below: V = h ; ; 8 i; and W = h, ; i: Is W a subspace of V? Explain your answer, showing all intermediate steps. Fall 99 points 8. If A; B; C and D are all ( ) matrices and E; F are ( ) matrices, is it then always true that " # " # " # A B E AE + BF =? C D F CE + DF points (i) True (ii) False

16 8. Let A bea(n n) square matrix with precisely one nonzero element in each row and precisely one nonzero element in each column of A. Does this guarantee that A has an inverse? points (i) True (ii) False 8. The equation below describes the row-reduction of a matrix A using elementary row operations:, {z } E, {z } E, {z } E,,8,, {z } A =,,, {z } U : Write downa( ) matrix L such that LU. points Ans L = 8. Does the subset W = f(x; y) j x; y; <; and x; y orx; y g i.e., the union of the rst and third quadrants of the (x; y)-plane <, constitute a subspace (over the real numbers) of the (x; y)-plane? (i) True (ii) False points 88. Is the matrix B =,

17 in row-reduced echelon-form? (In other words, does there exist a matrix A whichupon row-reduction to echelon form would yield B )? (i) Yes 89. Let A be the matrix Which of the following is true of A? (ii) No : points (a) If a solution to Ax = b exists, then that solution is unique (b) Ax = b always has at least one solution for every ( ) vector b (c) Neither of the above two statements is necessarily true points 9. The four vectors a, b, c and d lie in <. It is known that no vector in the set of vectors can be expressed as a linear combination of the remaining three vectors. Does it then follow that the set fa ;b ;c ;d g forms a set of four linearly independent vectors? (i) True (ii) False points 9. Does the vector x =,,, belong to the subspace of -dimensional space < spanned by the vectors a = and b =?

18 (i) Yes (ii) No points 9. Identify precisely all conditions on a; b; c; d under which the columns of a a a a a b b b a b c c a b c d are linearly independent. Before you begin, explain your approach in a brief sentence. 9. Use the Gauss-Jordan method to nd the inverse of the matrix,,, : points points 9. Consider the system of linear equations, x x x x = a b c d : The augmented matrix was row reduced to echelon form as shown below:, a b c d =), a b, a d, a c, b, a (a) Identify all values of a; b; c; d under which this system of linear equations will have at least one solution. points (b) Identify the free variables. 8 : point

19 (c) Find a particular solution for the case, a =;b=,;c=;d=. points (d) Find a basis for the nullspace of A. (A is the ( ) coecient matrix as usual). points (e) Find a basis for the rowspace of A. Spring Identify with a p all the answers that are correct: points If the rst and third rows of A are the same, so are the rst and third rows of AB If the rst and third rows of B are the same, so are the rst and third rows of AB If the square matrix C has an inverse, then (C ), = (C, ) points 9. The (n n) matrix M is given by M = " A B # where A and B are a pair of invertible (n n) matrices and denotes the (n n) all-zero matrix. Is it then true that " # A, M, = B, (i) Yes (ii) No points 9. The equation below describes the row-reduction of an n n matrix A using elementary row operations:, {z } E, {z } E {z } E Write downa( ) matrix L such that LU., {z } A =, {z } U : 9

20 points Ans L = 98. Identify geometrically, as clearly as you can, the subset of -dimensional (Euclidean) space < that corresponds to the column space of the marix : 99. The matrix A is an (m n) matrix with n>m. Then the equation Ax the rows of A are always dependent = always has at least one non-zero solution the columns of A are always dependent none of the above points Indicate with a p the statement(s) you think is (are) correct. points. The vector space V has a basis containing exactly vectors. Then (a) any vectors in V are linearly independent, (b) any vectors span V, (c) any vectors in V are linearly dependent, (d) None of the above is necessarily true. points. Let S be the subset of the (x,y)-plane (i.e., the subset of < ) given by Is S a subspace of the (x,y)-plane? S = f(x; y) j y =x +g:

21 points (i) Yes (ii) No. Reduce the matrix A below toechelon form. Use the echelon form to indicate with a ( p ) all the statements below that are true concerning the matrix A. (a) A has a left inverse (b) A has a right inverse,,,,,, (c) For every b <, Ax = b has a solution. (d) If Ax : = b has a solution, then that solution is unique. (e) none of the above is true of A. points. Elementary row-reduction using the Gaussian elimination proceedure, (no row interchanges were needed) of the (x) symmetric matrix A produced the upper triangular matrix U shown below: " # U = : (Thus, LU for some triangular matrix L with 's along the diagonal). Find the matrix A. points AnsA=. Use the Gauss-Jordan method to nd the inverse of the matrix :

22 Show all intermediate steps in your working clearly. points Ans. A, =. Extend the pair of vectors a = ; b = to a basis for <, i.e., nd a vector c such that fa;b;cg are a set of linearly independent vectors. points Before proceeding to solve the problem, explain in one or two sentences, your approach to the problem. points. Are the vectors v = ;v = ;v = and v = in < linearly independent? Show all intermediate steps and explain your answer. points. Let A be the (m n) matrix: Let B be the (n p) matrix B = y T y T y T n u T u T u T m Sp 99 = h v v v n i : = h z z z p i :

23 Write down a mathematical expresion for the matrix product AB in terms of (some or all of) the vectors u, v, y, z and their transposes. points 8. A rectangular (m n) matrix with m = n that has a left inverse also has a right inverse and vice-versa. (i) True (ii) False points 9. The matrix, is singular is nonsingular cannot say { could be either singular or nonsingular. points. The collection of all () matrices forms a vector space. Does the set of all nonsingular ( ) matrices form a subspace of this vector space? (i) True (ii) False points. In a certain system Ax = of homogeneous linear equations involving the four unknowns x ;x ;x and x, A is a ( ) matrix and the basic variables x and x are related as follows to the free variables x and x : x = x, x and x =x +x : Write down a basis for the nullspace of the matrix A. points

24 . Let V be the vector space of all continuous functions dened over the real line (the set of all real numbers). Identify the zero vector in this vector space. Are the continuous functions cos(x) and sin(x), (where x is in radians and varies over all the real numbers) linearly independent? (i) True (ii) False points. Let A be the matrix Determine the rank of A. : point Which of the following is true of A? If a solution to Ax = b exists, then that solution is unique Ax = b always has at least one solution Neither of the above two statements is necessarily true point. Express the matrix in the form P LDU where,, P isa() permutation matrix L isa() lower triangular matrix with 's along the diagonal D is a diagonal matrix U is a ( ) upper triangular matrix with 's along the diagonal, i.e., nd the matrices P, L, D and U.

25 8 points. Find necessary and sucient conditions on the components b ;b ;b and b under which the system of linear equations will have at least one solution.,,, x x x x =. Consider the system of linear equations Ax = b where c d : b b b b points Determine the set of all values of c and d for which this system of equations will have a unique solution.. Find a basis for the vector space W = < a ;a ;a ;a > points spanned by the ( ) vectors a =,, ; a =,, ; a = 8,, and a =,, : points Fall An (n n) lower triangular matrix has an inverse (a) i (if and only if) all the n diagonal elements equal

26 (b) i at least one diagonal element equals (c) i all the n diagonal elements are nonzero (d) always (e) none of the above points 9. Identify below with a ( p ) all the sets described below, that are vector spaces under the usual notion of addition and scalar multiplication: (a) the set fc + w j w W g; where c is a xed nonzero element of< n and W is a xed subspace of < n, (b) the set of all ( ) nonsingular matrices (c) the set of all polynomials of the form where c ;c ;c are real numbers, c + c X + c X + X (d) the set of all odd functions f : <!<, i.e., functions f(x) such that (e) none of the above f(,x) =,f(x) 8x <:. Indicate with a ( p ) all the statements below that are true concerning the matrix (a) A has a left inverse (b) A has a right inverse, (c) For every b <, Ax = b has a solution. (d) If Ax : = b has a solution, then that solution is unique. (e) none of the above is true of A. points

27 points. Upon row-reduction, the (m n) matrix A yields the row-reduced echelon matrix U. Indicate below with a ( p ) all the correct answers concerning A, U and A T A that you can spot: (a) A and U have the same rowspace, (b) A and U have the same column space, (c) A and A T A have the same rank, (d) A and A T A have the same column space, points. Consider the linear transformation L that projects each point (x; y) in the (x; y)-plane to the nearest point L( (x; y) ) on the straight line given by y = x. For example the point P in the gure above is projected " # onto the point Q lying on the x straight line x = y. Find a matrix A such that A is the vector whose coordinates y are the coordinates of the point L( (x; y) ) for all points (x; y) in the plane. You may assume throughout, that the basis in use is the standard basis for <. points. Identify a vector z in < of the form [z ;z ; ; ] T which makes a angle with the vector [ ] T lying in < and which moreover, has unit length, i.e., z + z =. points

28 . Given the matrices and B = A(A T A), A T = = = = = nd the projection of the vector [ ] T onto the column space of the matrix B. ; points. The Gram-Schmidt procedure, applied to the columns a ;a ;a ;a of the matrix A given by h i a a a a = ;, resulted in the set q ;q ;q ;q of orthonormal vectors which are the columns of the orthogonal matrix Q given by Q = h q q q q i = = = = = =,= =,= = =,=,= =,=,= = In the associated QR decomposition of the matrix A, determine the elements belonging to the third column of R, i.e., if determine r ;r and r. Q r r r r r r r r r r ; : points. Express the matrix in the form P LDU where, 8 P isa( ) permutation matrix 8

29 L isa() lower triangular matrix with 's along the diagonal, U is a ( ) upper triangular matrix with 's along the diagonal, D is a ( ) diagonal matrix, i.e., identify the matrices P, L, D and U. 8 points. Given the ( ) matrix row-reduce A to an echelon matrix U., 8 ; points Next, use U to identify a basis for (a) the column space of A, points (b) the rowspace of A points (c) and the nullspace of A. points 8. Let h ;h ;h ;h be the linearly independent column vectors of the matrix (a) Compute H T H. H = h h h h h i = =,=,=,= : (b) Express [ ; ; ; ] T as a linear combination of the vectors h ;h ;h ;h. points points 9

30 (c) Project [ ; ; ; ] T onto the space spanned by h and h, i.e., onto the column space of the matrix,=,= : Fall 988 points 9. For every pair of square matrices A and B (of common size (nxn)), it is always true that (a) (AB) k = A k B k (b) (A ), =(A, ) wheneveraisinvertible. Indicate with a ( p ) the statement(s) you think is(are) correct.. In the equation AB = C shown below, A and C have been partitioned into blocks (submatrices). Show how you would similarly partition B so as to express the blocks of C in terms of the blocks of A (and the appropriate blocks of B). a a a a a a a a a a a a b b b b b b b b b b b b = c c c c c c c c c. A square (x) matrix A has been factored into the product of a lower (L) and an upper (U) triangular matrix i.e., LU: Ignoring additions and subtractions and counting only either multiplications or divisions as operations, roughly how many operations are now required in general to solve an equation of the form: LUx = b for a given right hand side vector b? roughly, :

31 roughly roughly roughly roughly Mark the single most appropriate answer.. Elementary row-reduction using the Gaussian elimination proceedure, (no row interchanges were needed) of the (x) symmetric matrix A produced the upper triangular matrix U shown below: " # U = : (Thus, LU for some triangular matrix L with 's along the diagonal). Find the matrix A. AnsA=. Let S be the subset of the (x,y)-plane given by Is S a subspace of the (x,y)-plane? S = f(x; y)=y =x +g: (i)yes (ii)no. For what values of the constants a,b and c does the set of equations given below in augmented matrix form have at least one solution? Ans... a, b, a c, a

32 . Let A be an (mxn) matrix. Consider the linear transformation f(:) :< n!< m given by f(x) =Ax: (a) What must be the rank of A to ensure that f(.) is a one-one mapping, i.e., to ensure that for all pairs x, y Ans... f(x) =f(y) =) x = y? (b) What must be the rank of A to ensure that f(.) is onto, i.e., to ensure that for every y in < m, there is at least one x in < n such that Ans... f(x) =y?. Of the vectors a, b and c lying in -dimensional Euclidean space (i.e., < ), only a and b lie in the (x,y)-plane. Also it is known that the vectors a and b do not lie on the same straight line passing through the origin. Do these conditions guarantee that a, b and c form a basis for <? (i) Yes (ii) No. (iii)cannot say.. A is an (mxn) matrix and x and x distinct vectors belonging to the rowspace of A. Is it possible that Ax = Ax? (Hint: Consider A(x, x ).) (i) Yes (ii)no. 8. Find the angle between the vectors x ==

33 and lying in <. y = Ans Given solve LU = Ax = 8,,,, without multiplying L and U to obtain A rst. Show all intermediate steps clearly. ; pts. Determine a basis for and the dimension of the columnspace of the matrix A given by:.,,,,,, pts. Let P(d) denote the vector space of all polynomials having real coecients and of degree d, i.e., P (d) =fa + a X + a X + :::: + a d X d =all a i realg: Since integration of a polynomial of degree yields a polynomial of degree, the integration operator may be viewed as a linear transformation mapping from P() to P().

34 (a) Write down a matrix S that represents this linear transformation. Identify clearly your choice of basis for P() and P(). (b) What is the nullspace of this matrix? Sp 99. The matrix A as well as it's inverse A, are given below: The matrix,,,, and A, = B =,,,,,8,,,9 is obtained by interchanging the rst two rows of A. Write down the inverse of B. ANS : pts pts points B, =. In the n-dimensional space < n, let addition and scalar multiplication be dened as follows: x y = x, y ; and c x =,cx for any pair of vectors x, y < n and any real number (scalar) c. Under this denition of addition () and scalar multiplication (), is < n avector space? Justify your answer briey. points

35 ANS : Justication:. Determine the dimension of the nullspace of the ( ) matrix,,, : points ANS The dimension of the nullspace is.. If the n columns of an n n matrix A are linearly independent, the same is true of the columns of the matrix A. points (i) True (ii) False. Let W denote the subspace of ( ) matrices spanned by the elementary matrices E i;j, with ones on the diagonal and at most one nonzero entry below. What is the dimension of W? points ANS The dimension of W =.. Let A be an (m n) matrix. In least-squares approximation, given an (m ) column vector b, one determines an unique vector Ax that minimizes jjax, b jj : Is the mapping from b to Ax a linear transformation?

36 points (i) Yes (ii) No 8. The vector space V has a basis containing exactly vectors. Then (a) any vectors in V are linearly independent, (b) any vectors span V, (c) any vectors in V are linearly dependent, (d) None of the above is necessarily true. points 9. Exhibit a pair (x;y), of distinct, non-zero vectors belonging to < satisfying jx t y j = jjx jj jjy jj: points ANS x = ; and y = :. Let b be an (m) column vector and A an (mn) matrix. Let x denote the (n) vector that minimizes jjax, b jj : In not more than two lines (you may use equations if you wish) identify precisely, the set of all vectors b for which the corresponding minimizing vector x =. ANS points

37 . Write down a basis for W? where W is the nullspace of the matrix : points ANS A basis is. In this question, we will identify points (x; y; z) in -dimensional space < with the vectors [x yz] t. Consider the linear transformation T : <! < that rotates every point in -dimensional space 9 about the z-axis in the direction shown in the gure above. (a) Write down the transformed coordinates (a; b; c) (i.e., after rotation) of the point (; ; ). point (b) Determine a matrix that carries out the linear transformation T. (Note that the standard basis fe ;e ;e g is used to provide a coordinate representation for both domain as well as range of T ). points

38 (c) Is the answer to part (b) unique? point (a) ANS (a; b; c) = : (b) ANS The matrix of the linear transformation is : (c) ANS Answer to the uniqueness question is:. (a) Reduce the matrix to an (row-reduced) echelon matrix U.,,,,,, points (b) Hence determine a basis B row for the rowspace Row(A) aswell as a basis B col for the column space C(A) of A. points ANS (a) The echelon matrix U is given by: 8

39 U = ANS (b) B row = B col =. (a) State in the form of an equation, the orthogonality principle as it applies to leastsquares approximation. (b) Given the QR decomposition p p nd the vector Ax that minimizes jjax b = Explain your approach in a brief sentence. (a) ANS The orthogonality principle: (b) ANS 9 " p p, b jj when : # ; points points

40 My approach is the following: The minimizing vector Ax is:. A isa( ) matrix and Write down the matrix A if AB = B = h 9 i + Sp 99, 9 8 : h i, h 8 i : points. For what values of the constants a,b and c does the set of equations given below in augmented matrix form have at least one solution? Ans... 9 a b + a c, a. The matrix A as well as it's inverse A, are given below: The matrix,,, B = and A, =,,,,,,,,,,,, is obtained by interchanging the rst two rows of A. Write down the inverse of B. ANS : points

41 B, = 8. Let V be the set of all positive real numbers, with addition () and multiplication () dened by: x y = xy c x = x c where c is any scalar (i.e., any real number). Is V with these operations, a vector space? (i) True (ii) False 9. The collection of all () matrices forms a vector space. Does the set of all nonsingular ( ) matrices form a subspace of this vector space? (i) True (ii) False points. If the n columns of an n n matrix A are linearly independent, the same is true of the columns of the matrix A. points (i) True (ii) False. Let W denote the subspace of ( ) matrices spanned by the elementary matrices E i;j, with ones on the diagonal and at most one nonzero entry below. What is the dimension of W? points

42 ANS The dimension of W =.. A is an (m n) matrix with n>m. Then (a) the equation Ax =always has at least one nonzero solution (b) the rows of A are linearly dependent (c) the columns of A are linearly dependent (d) none of the above Indicate with a ( p ) the statement(s) you think is (are) correct. points. In a certain system Ax = of homogeneous linear equations involving the four unknowns x ;x ;x and x, A is a ( ) matrix and the basic variables x and x are related as follows to the free variables x and x : x = x + x and x =x, x : Write down a basis for the nullspace of the matrix A. points. Express the matrix in the form P LDW where,, 9 P isa() permutation matrix L isa() lower triangular matrix with 's along the diagonal D is a diagonal matrix W is a ( ) upper triangular matrix with 's along the diagonal, i.e., nd the matrices P, L, D and W. 8 points

43 . Find a basis for the vector space W = < a ;a ;a ;a > spanned by the ( ) vectors a =, ; a = 8, ; a =,,, and a =,,, : points. Does the system of linear equations x, x +x = x + x = x, x +x = : have a solution? If so, describe explicitly, all solutions. points

44 PRIOR FINAL EXAMS Fall 99. Let W and W be subspaces of < m. Let f ; ;:::; k g and f ; ;:::; l g be a basis for W and W respectively. You may assume k l and m > (k + l): Let A be the (m (k + l)) matrix given by h ::: k ::: l i : In terms of m; k; l what are the maximum and minimum possible dimensions of the nullspace of A? ANS The dimension of the nullspace can range between: points 8. The distinct eigenvalues of a ( ) complex matrix A are,j; j; +; and, where as usual, j = p (,). What is the least value of the integer k such that where I is the ( ) identity matrix? ANS least value of k = A k = I points 9. Let P be an (n n) (with n ) projection matrix, i.e., P is a non-zero matrix that satises P = P T = P : Then P has precisely two distinct eigen values, namely and. (i)true (ii)false points

45 . Let a be an arbitrary complex number. Let A be the ( ) matrix " # a a a : a Is A always guaranteed to have a pair of orthonormal vectors as eigenvectors? (i)true (ii)false points. Let A be a ( ) matrix whose determinant equals. Let r ;r ;r be the ( ) row vectors corresponding to the rows of A, i.e., What is the determinant of the matrix ANS det(b) = B = r r r : r + r r + r r + r. Identify any two distinct points lying on the straight line given by the equation: ANS The two points are: x y 8 = :? points points. The matrix is ",, #

46 (a) positive denite (b) positive semidenite (c) negative denite (d) none of the above points. Let A be a positive denite Hermitian matrix. Then for every vector x having length jjx jj =, the value of the quadratic form cannot exceed the trace of the matrix A. x H Ax (i)true (ii)false points. The diagonal form of a certain diagonalizable complex matrix A (i.e., thus for some S, SS, ) is of the form: where j = p (,). Then =, +j ; (a) A could be a real-valued matrix; (b) A could be a Hermitian matrix; (c) A could be a unitary matrix; (d) none of the above is possible. points. Every real, square matrix A is similar to its transpose A T. (i)true (ii)false

47 points. Rewrite the scalar recursion as a vector recursion of the form f k = f k,, f k, + f k, ; k ; a k = Aa k, for a suitably chosen matrix A and vector a k. Clearly identify the components of both matrix as well as vector. points 8. Consider the dierential equation du(t) dt Which of the following is then true: = " # u(t): (a) for some initial conditions u(), the output u(t),! as t,! ; (b) for some initial conditions u(), the output u(t),! as t,! ; (c) no matter what the initial conditions are, the output will neither decrease to zero nor remain unbounded as t,! ; (d) none of the above points 9. Let A be the matrix : (a) Write down the characteristic polynomial f(x) of A. points ANS f(x) = (b) What is the minimum polynomial p(x) of A?

48 points ANS p(x) = (c) Is A diagonalizable? Explain your answer in a single sentence. points ANS 8. The matrices A and B,, share the same characteristic polynomial as well as the same minimum polynomial and B = f(x) = (x, ) p(x) = (x, ) : 9,,, 9,,, Determine the Jordan canonical forms J A and J B of the two matrices. points Before proceeding to the computations, outline your approach in a brief sentence. ANS J ; J B = My approach: 8

49 8. Given the quadratic x, x x + x ; it is known that there exists a linear change of variables of the form " # " # y x = A y x which will allow the quadratic form to be written in the form ay + by for some real numbers a and b. Find the matrix A as well as the real numbers a and b. Before proceeding to the computations, outline your approach in a brief sentence. points ANS a = ; b = 8. Let My approach: extra working space Q Q T be the singular value decomposition of the ( ) matrix A. The matrix A has rank and the matrices in the decomposition are given by Q = g g g g g g g g g ; = ; Q = h h h h h h h h h h h h h h h h In terms of the elements of the matrices Q, and Q, write down an orthonormal basis for 9 :

50 (a) the columnspace of A; points (b) the nullspace of A. points Finally, determine the eigenvalues of A t A. point F Let W be the set of all polynomials of degree, whose rst coecients are equal, i.e., W = fc + c X + c X + c X + c X + c X + c X j c = c = c = c g: Let W be the set of all polynomials of degree, whose last coecients are equal, i.e., W = fc + c X + c X + c X + c X + c X + c X j c = c = c = c = c g: (All coecients are real). Find the dimension of the sumspace W + W as a vector space over the real numbers. points 8. Let A bea( ) matrix whose determinant equals. Let r, r and r be the ( ) row vectors corresponding to the rows of A, i.e., Determine the determinant of the matrix B = r r r : r, r r + r r, r : points

51 8. In this question, the underlying eld of scalars is the set of all complex numbers C. Let A be a matrix with complex components. As vector spaces over C, is it true that the rowspace of A is the orthogonal complement of the nullspace of A, i.e., ( Row(A) )? = N (A)? points (i) True (ii) False 8. Again, in this question, the underlying eld of scalars is the set of all complex numbers. Then, the distance between any two vectors x; y C n, is dened by d(x;y) = jjx, yjj = q (x, y) H (x, y): Does an unitary matrix U always preserve distances, i.e., is it true that for all x, y C n, d(x;y) = d(ux;uy)? points (i) True (ii) False (iii) Cannot Say. 8. Let a be an arbitrary complex number. Let A be the ( ) matrix " # a a a a where `() ' denotes the complex conjugate. Is A a normal matrix? points (i) True (ii) False (iii) Cannot Say. 88. The diagonal form of a certain diagonalizable complex matrix A is given by =, +i ; where i = p,. (Thus, SS, for some S). Then

52 (a) A could be a real-valued matrix (b) A could be a Hermitian matrix (c) A could be an unitary matrix (d) None of the above is possible. points 89. Let P be a (n n) n> projection matrix, P = [], P = I, where [] and I are the (n n) all-zero and identity matrices respectively. Then (a) P has at most two distinct eigen values and these are and, (b) P has exactly two distinct eigen values and these are and, (c) P has at least two distinct eigen values and these are and. Indicate the single answer that you feel is most appropriate. points 9. A, S, C and are all (n n) matrices. S and C are nonsingular and ; are diagonal matrices. It turns out that A can be expressed in the form Then S S, = C C H : (a) the column vectors of S must necessarily be a set of linearly independent eigen vectors of A. (b) the column vectors of C must necessarily be a set of linearly independent eigen vectors for A. (c) none of the above is necessarily true. 9. Let U be an (n n) unitary matrix. Then for every complex vector x C n is always real. x H Ux points (i) True (ii) False (iii) Cannot say.

53 9. Let x be the ( ) real vector [x x x ] T and consider the real quadratic form points f(x) = x + x + x, x x, 8x x in the three components of x. Express the quadratic form as f(x) = x T Ax for a suitably chosen symmetric matrix A and identify A. points 9. In this question, we will identify points (x; y; z) in -dimensional space < with the vectors [x yz] T. (a) Let A and B be two dierent bases for < given by and B = 8 >< >: 8 >< >: ; ; Let the matrices [I] AB and [I] BA correspond to a change of coordinates from A to B and vice-versa, i.e., [I] AB [x] [x] B and ; ; [I] BA [x] B = [x] A for all vectors x belonging to <. Determine either [I] AB or [I] BA, but make clear in your working, as to which of the two matrices you have chosen to determine. 9 >= >; 9 >= >; : points (b) Consider the linear transformation T : <! < that rotates every point in -dimensional space 9 about the z-axis in the direction shown in the gure above. Write down the transformed coordinates (a; b; c) (i.e., after rotation) of the point (; ; ). point

54 (c) Determine the matrix [T ] BB that carries out the linear transformation when the vectors in both the domain and range of the transformation T are represented with respect to the basis B, i.e., [T ] BB : [x] B! [T (x)] B for all vectors x in <, where T (x) denotes the image of x under the linear transformation T. Show all of your working clearly. (See previous page for the basis B). 9. Let A and B be the matrices and B = ", " # # : points Determine for each matrix, whether or not the matrix is diagonalizable. Show clearly, your working. 9. Find the determinants of the upper-left submatrices of the matrix A given by Is A positive-denite?,,,, 9. Let a>b>c> be real numbers. Show that the minimum value of : ay + by + cy subject to y + y + y = where y ;y and y are real numbers, equals c. points points points

55 9. What is the minimum value of the quadratic form x T Ax where,,,, subject to x T x = x + x + x =, where x ;x and x are real numbers. Explain your working clearly. (Hint: Use QQ T, for some orthogonal matrix Q. Note that Q preserves lengths). Sp Let Q be an (n n) real, orthogonal matrix. Then for every vector x < n, jjq xjj = jj x jj : : points (i) True (ii) False Let A be an (n n) matrix that satises jja xjj = jj x jj for every vector x < n. Then A must be an orthogonal matrix. (i) True (ii) False points 99. Let A be an (m n), real matrix. Let b be an arbitrary vector in < m. Consider the set S = fa x j x is a solution to the equation A T Ax = A T b g : How many distinct elements does S contain? (a) S always contains just one distinct element (b) S always contains an innite number of distinct elements (c) the number of distinct elements depends upon the particular b

56 (d) none of the above is necessarily true points. The vectors a and a in < are orthogonal. The projection of a certain vector b (also in < )onto a turns out to be the vector. Let The projection of b onto the subspace : W = < a ;a > of < spanned by both a and a turns out to be the vector Determine the projection of b onto a. : f ; ; g and B = f ; g points be bases for the orthogonal subspaces W A and W B respectively. W A and W B are subspaces of <.How is the columnspace of the matrix h ; ; ; ; i related to W A and W B? What is the rank of A? points. A isa( ) matrix whose (i; j) th element is denoted as usual by a i;j. Let A i;j denote the cofactor of A associated with a i;j. Find a ( )matrix B whose determinant has the following cofactor expansion det(b) = xa + ya + za points

57 . The matrix A is a ( ) real matrix having only two distinct eigen values, namely, and. What are the possible values of the determinant ofa?. The eigenvalues of a certain complex ( ) matrix A are ; j;,;,j where as usual, j = p,. What is the least value of the integer n such that where I is the ( ) identity matrix? A n = I;. Consider the sequence of integers that satisfy the recursion G k = G k,, G k, + G k, ; 8k : points points The initial conditions are G =,G = and G =. Rewrite this scalar recursion as a single-step ( ) vector recursion. points. Let A be an (n n) Hermitian matrix. Then for every arbitrary complex vector x C n (where C denotes the set of all complex numbers), x H Ax is always real. (i) True (ii) False. Let W be the subspace of < spanned by the orthogonal vectors Express the vector u = u = z =, : points as the sum of two vectors z and z, where z W and z is perpendicular to all the vectors in W. Explain your intended procedure clearly before you begin.

58 8 points 8. Let A and B denote the two bases for < given by " # " # " f ; g and B = f # ; ", # g: Note that A is the standard basis for <. Let S be the matrix S = ",, If coordinate representation with respect to the basis B is used instead (of the standard basis A), determine the representation S BB of S with respect to the basis B. # : 8 points (As intermediate steps, determine and identify clearly, the change-of-basis matrices T AB (from A!B) and T BA (from B!A).) 9. Find the limiting values of y k and z k (i.e., the values as k!)if y k+ = :8y k + :z k y = () z k+ = :y k + :z k z = () Fall 99 8 points. Let W be the set of all polynomials of degree, whose rst coecients are equal, i.e., W = fc + c X + c X + c X + c X + c X + c X j c = c = c = c g: Let W be the set of all polynomials of degree, whose last coecients are equal, i.e., W = fc + c X + c X + c X + c X + c X + c X j c = c = c = c = c g: (All coecients are real). Find the dimension of the sumspace W + W as a vector space over the real numbers. 8 points

59 . Let A bea() matrix whose determinant equals. Let r, r and r be the ( ) row vectors corresponding to the rows of A, i.e., Determine the determinant of the matrix B = r r r : r, r r + r r, r : points. In this question, the underlying eld of scalars is the set of all complex numbers C. Let A be a matrix with complex components. As vector spaces over C, is it true that the rowspace of A is the orthogonal complement of the nullspace of A, i.e., ( Row(A) )? = N (A)? points (i) True (ii) False. Again, in this question, the underlying eld of scalars is the set of all complex numbers. Then, the distance between any two vectors x; y C n, is dened by d(x;y) = jjx, yjj = q (x, y) H (x, y): Does an unitary matrix U always preserve distances, i.e., is it true that for all x, y C n, d(x;y) = d(ux;uy)? points (i) True (ii) False (iii) Cannot Say.. Let a be an arbitrary complex number. Let A be the ( ) matrix " # a a a a where `() ' denotes the complex conjugate. Is A a normal matrix? 9

60 points (i) True (ii) False (iii) Cannot Say.. The diagonal form of a certain diagonalizable complex matrix A is given by =, +i ; where i = p,. (Thus, SS, for some S). Then (a) A could be a real-valued matrix (b) A could be a Hermitian matrix (c) A could be an unitary matrix (d) None of the above is possible. points. Let P be a (n n) n> projection matrix, P = [], P = I, where [] and I are the (n n) all-zero and identity matrices respectively. Then (a) P has at most two distinct eigen values and these are and, (b) P has exactly two distinct eigen values and these are and, (c) P has at least two distinct eigen values and these are and. Indicate the single answer that you feel is most appropriate. points. A, S, C and are all (n n) matrices. S and C are nonsingular and ; are diagonal matrices. It turns out that A can be expressed in the form Then S S, = C C H : (a) the column vectors of S must necessarily be a set of linearly independent eigen vectors of A. (b) the column vectors of C must necessarily be a set of linearly independent eigen vectors for A.

61 (c) none of the above is necessarily true. points 8. Let U be an (n n) unitary matrix. Then for every complex vector x C n is always real. x H Ux (i) True (ii) False (iii) Cannot say. 9. Let x be the ( ) real vector [x x x ] T and consider the real quadratic form points f(x) = x + x + x, x x, 8x x in the three components of x. Express the quadratic form as f(x) = x T Ax for a suitably chosen symmetric matrix A and identify A. Sp 99 points. Let W be the set of all polynomials of degree, whose rst coecients are equal, i.e., W = fc + c X + c X + c X + c X + c X + c X j c = c = c g: Let W be the set of all polynomials of degree, whose last coecients are equal, i.e., W = fc + c X + c X + c X + c X + c X + c X j c = c = c = c = c g: (All coecients are real). Find the dimension of the sumspace W + W as a vector space over the real numbers. points