Extra Problems for Math 5 Linear Algebra I Find the vector AB and illustrate with a picture if A = (,) and B = (,4) Find B, given A = (,4) and [ AB = A = (,4) and [ AB = 8 If possible, express x = 7 as a scalar multiple of u = 8, as a scalar multiple 4 6 7 of v =, and as a scalar multiple of w = Justify your answers 4 7 4 4 Given u =, u = parallel vectors, u = 6 6, u 4 =, u 5 = 5 Find the indicated vectors [ [ (a) 4 + (b) 4 5 6 Suppose u = [ [ a, v = 5 6 b and w = [ Find a and b if u v = w 5, list all pairs of 7 Suppose x, y, u, and v are vectors Express u and v in terms of x and y given x y = u and x + y = v [ [ 4 8 Given u = and v =, find u v and illustrate with a picture like that in Figure 6 9 Shown at the right are two nonparallel vectors u and v and four other vectors w,w,w,w 4 Reproduce u, v, and w in a picture by themselves and exhibit w as the diagonal of a parallelogram with sides parallel to u and v Guess (approximate) values of a and b so that w = au + bv w w u v w 4 w Express [ 7 7 Is it possible to express as a linear combination of [ [ and [ as a linear combination of [ and [? Explain c MMVIII Edgar G Goodaire, Dept of Mathematics and Statistics, Memorial University of Newfoundland
Extra Problems 4 Let v =, v = 5, and v = 6 Is a linear combination of v, v, v? Is it possible to express the zero vector as a linear combination of more than one way? Justify your answer [ 4 and [ 8 in 4 Suppose u and v are vectors Show that any linear combination of u and v is a linear combination of u and v 5 Let O(,), A(,), B(4,4), C(, ), D(4,), E(6,), F(, 4), G(6,), H(8,) be the indicated points in the xy-plane Let u = OA be the vector pictured by the arrow from O to A and v = OC the vector pictured by the arrow from O to C Find u and v Then express the vector [ OD from to D as a combination of u and v For example, 6 OE = = u + v A 6 Suppose ABC is a triangle Let D be the midpoint of AB and E be the midpoint of AC Use vectors to show that DE is parallel to BC and half its length B D E C 7 Let A = (, ), B = (4, ) and C = (, ) be three points in the Euclidean plane Find a fourth point D such that the A, B, C and D are the vertices of a parallelogram and justify your answer Is the answer unique? 8 Prove that vector addition is closed and commutative and every vector u = negative u which has the property that u + ( u) = ( u) + u = [ x y has a 9 Show that any linear combination of and is also a linear combination of 6 and Explain how one could use vectors to show that three points X, Y, and Z are collinear (Points are collinear if they lie on a line) If u and v are parallel, show that some nontrivial linear combination of u and v is the zero vector (Nontrivial means that u + v = doesn t count!)
Extra Problems Let u = and v = Find a unit vector in the direction of u and a vector of 4 length in the direction opposite to v Susie is asked to find the length of the vector u = 4 7 She proceeds like this 5 8 7 4 u = 7, so u = ( 8 7 ) 4 + ( 7 ) + ( 7 ) = 487 7 Do you like this approach? 4 Find the angle between each of the following pairs of vectors Give your answers in radians and in degrees If it is necessary to approximate, give radians to two decimal places of accuracy and degrees to the nearest degree (a) u = [ 4, v = [ 4 (b) u = [ 4, v = [ 5 Find u v given u =, v =, and the angle between u and v is 45 6 Can u v = 7 if u = and v =? Justify your answer 7 Let u = 4, v =, and w = (a) Find u, u + v and w w (b) Find a unit vector in the same direction as u (c) Find a vector of norm 4 in the direction opposite to v 8 Let u and v be vectors of lengths and 5 respectively and suppose that u v = 8 Find (u v) (u v) and u + v [ [ 9 Let u = and v = Find all numbers k such that u + kv has norm If a nonzero vector u is orthogonal to another vector v and k is a scalar, show that u + kv is not orthogonal to u Given the points A(,) and B(,) in the plane, find all points C such that A, B, and C are the vertices of a right angle triangle with right angle at A, and AC of length Find a point D such that (,), (, ), (4, ) and D are the vertices of a square and justify your answer
4 Extra Problems Let u = 4, w = 5 and v = 7 4 Show that u is orthogonal to v and to w 4 Assuming all vectors -dimensional, prove that the dot product is commutative and that v v = if and only if v = [ [ 5 Verify the Cauchy Schwarz inequality for u = and v = 5 6 Use the Cauchy Schwarz inequality to prove that (acos θ + bsin θ) a + b for all a,b R 7 Let a,b,c,d be real numbers Use the Cauchy Schwarz inequality to prove that (ac+ bd) (a + b )(c + d ) 8 Suppose vectors u and v are represented by arrows OA and OB in the plane, as shown Find a formula for a vector whose arrow bisects angle AOB 9 Let P be a parallelogram with sides representing the vectors u and v as shown Express the diagonals AC and u O B v B C u A D v DB of P in terms of u and v 4 Express the plane with equation x y + z = as the set of all linear combinations of two nonparallel vectors (There are many possibilities!) 4 Find the equation of each of the plane parallel to the plane with equation 4x+y z = 7 and passing through the point (,,) 4 Verify that u v = (v u) given u = and v = 4 4 Given u = and v =, find the cross product u v, then verify that this vector is perpendicular to u and to v and that v u = (u v) 4 44 Given u =, v =, and w =, compute (u v) w and u (v w) Are you surprised at the results? 45 Let a and b denote arbitrary scalars Find au + bv given u = and v = 4 Then find the equation of the plane that consists of all such linear combinations of u and v A?
Extra Problems 5 46 Find two vectors perpendicular to both u = and v = 47 Find a vector of length 5 that is perpendicular to both u = 48 Verify that u v = u v sin θ with u = and v = and v = 5 49 Find the equation of the plane passing through A, B, and C, given A(,,), B(,,5), C(,, 4) x 5 Find the equation of the plane containing P(,, ) and the line with equation y = z 4 + t 5 Find equations of [ the form [ ax [ + by = c for the lines in the Euclidean plane with the x vector equation = + t y 5 Find vector equations for the line in the Euclidean plane whose equation is y = x 5 The intersection of two planes is, in general, a line (a) Find three points on the line that is the intersection of the planes with equations x + y + z = 5 and x y + z = (b) Find a vector equation for the line in 5(a) x 54 Determine, with a simple reason, whether the line with equation y = +t 5 z and the plane with the equation x y + z = 4 intersect Find any points of intersection 55 Let π be the plane with equation x y + 5z = Find a vector perpendicular to π 56 Let A(,, ), B(,, ), C(, 5,), D(, 4,) be the indicated points of - space (a) Show that A,B,C,D are the vertices of a parallelogram and find the area of this figure (b) Find the area of triangle ABC
6 Extra Problems 57 Find the area of the triangle in the xy-plane with vertices A(, ), B(4, 4), C(5, ) 58 Suppose a determinant a b [ [ c d = Show that one of the vectors c a, is a d b scalar multiple of the other There are four cases to consider, depending on whether or not a = or b = x x 59 Determine whether or not the lines with equations y = + t and y = z z 4 +t intersect Find any point of intersection 5 6 Let l and l be the lines with the equations x l : y = + t ; l : z x y = + t z a b Given that these lines intersect at right angles, find the values of a and b 6 Find the equation of the plane each point of which is equidistant from the points P(,,) and Q(,, ) 6 When does the projection of u on a nonzero vector v have a direction opposite that of v? 6 Let u = and v = Determine whether w = is in the plane spanned by u and v 64 In each case below, find the projection of u on v and the projection of v on u (a) u = and v = (b) u = and v = 4 65 Find the distance from point P(,,) to the line l with equation t x y = z + 66 Find the distance from P(,,) to the plane π with equation 5x y + z = Then find the point of π closest to P 67 Find two orthogonal vectors that span each of the given planes:
Extra Problems 7 (a) the plane spanned by u = and v = 4 (b) the plane with equation x y + 4z = 68 (a) Find two orthogonal vectors in the plane π with equation x y + z = (b) Use your answer to part (a) to find the projection of w = on π 69 (a) What is the projection of w = onto the plane π with equation x+y z =? x (b) What is the projection of w = y onto π? z 7 Find the projection of w = on the plane π whose equation is x + y z = 7 Suppose a vector w is a linear combination of nonzero orthogonal vectors e and f Prove that w = w e e e e+ w f f f f f w f f f f w e e e e e x x 7 (a) The lines with vector equations y = + t and y = z 4 z are parallel Why? (b) Find the distance between the lines in (a) w 4 + t 7 Most theorems in mathematics can be stated in the form If P, then Q, symbolically P = Q (Read implies for = ) The converse of this statement is Q = P: If Q, then P What is the converse of Theorem 5? 74 Don has a theory that to find the projection of a vector w on a plane π (through the origin), one could project w on a normal n to the plane, and take proj π w = w proj n w (a) Is Don s theory correct? Explain by applying the definition of projection onto a plane (b) Illustrate Don s theory using the plane with equation x y+z = and w = 6 6
8 Extra Problems 75 (The distance between skew lines) Let l and l be the lines with equations x 4 x l : y = + t, l : y = + t 6 z z 7 (a) Show that l and l are not parallel and that they do not intersect (Such lines are called skew) (b) Find the equation of the plane that contains l and is parallel to l (c) Use the result of (b) to find the (shortest) distance between l and l 76 Let u =, v =, x =, and y = Find each of the following, if possible (a) x y + u (b) y 77 Find a and b, if possible, given a b a b = b 6 b 8 c a 78 Let u = b, v =, x =, and y = a 4 4 4 5 8 Find a, b, and c, if possible, given u v = y 79 Find a unit vector that has the same direction as u = 8 Let u = and v = Find all scalars c with the property that cu = 8 8 Find the angle (to the nearest tenth of a radian and to the nearest degree) between u = and v = 8 Suppose u is a vector of length 6 and v is a unit vector orthogonal to u + v Find u v 8 Suppose u, v, x, and y are unit vectors such that v x =, u and y are orthogonal, the angle between v and y is 45, and the angle between u and x is 6 Find (u+v) (x y)
Extra Problems 9 84 Determine, with justification, whether v = is a linear combination of v 4 = 5, 4 5 4 5 v = 4, v =, v 5 4 =, and v 5 = 5 4 5 4 85 Verify the Cauchy Schwarz and triangle inequalities for u and v = 86 Determine whether v =, v =, and v =, v 4 = are linearly 4 4 independent or linearly dependent 87 Why have generations of linear algebra students refused to memorize the definitions of linear independence and linear dependence? 88 Suppose you were asked to prove that vectors v,v,,v k are linearly independent What would be the first line of your proof? What would be the last line? x y 89 Find x and y, if possible, given x y x + y y x + y = x + y [ [ a + x 9 Find x, y, a, and b if possible, given = y + b 9 Let A = [ x y [, B = z 4 5 Find x, y, and z, if possible, so that (a) A B = C (b) AB = C [ 7, C = 8 a x a + x [ 7, and D = 9 What is the matrix A for which a ij = j? [ 9 Find A+B, A B, and A B given A = [ 94 Find AB and BA (if defined) given A = 4 4 6 8 [ and B = and B = 4 [ 4 6
Extra Problems [ x u a b c 95 Let A = and B = y v What are the (,) and (,) entries of d e f z w AB? [ [ [ 96 Write + 4 in the form Ax, for a suitable matrix A and vector x [This 8 question tests understanding of the very important fact expressed in 6 97 Express the system x y = x + 5y = 7 x + y = x 5y = in the form Ax = b 98 The parabola with equation y = ax + bx + c passes through the points (,4) and a (,8) Write a matrix equation whose solution is the vector b c 99 Complete the following sentence: A linear combination of the columns of a matrix A is 4 9 Let A = 4 5 6, x =, and b = Given that Ax = b, write b as a 7 8 9 7 4 linear combination of the columns of A [ [ Express b = as a linear combination of the columns of A = Then find a vector x so that Ax = b Let I = be the identity matrix Express x = combination of the columns of I 5 as a linear 47 Suppose x, x, and x are three-dimensional vectors and A is a matrix such that x x x Ax =, Ax =, Ax = Let X = be the matrix with columns x, x, x What is AX? 4 Let A = and D = explain your reasoning 7 8 5 Find AD without calculation and
Extra Problems 5 Let A = [ 4 [ 4, B = 4 [, C = 5 Compute AB, AC, AB + AC, B + C, and A(B + C) 6 Let A be a square matrix Does the equation (A [ I)(A + I) = imply A = I or A = I? Answer by considering the matrix A = 7 If A is a matrix and x is a vector such that x+ax is a vector, what can you say about the size of A (and why)? 8 If A and B are m n matrices and Ax = Bx for each vector x in R n, show that A = B 9 Let x = x x x n be nonzero and let y be any vector in Rm Prove that there exists an m n matrix A such that Ax = y Determine whether or not A = are inverses Given A = Let A = [ 5 [ 7 9 (a) Compute AB and BA and B = 5 5 find A and use this matrix to solve the system and B = (b) Is A invertible? Explain [ Let A = and B = A T B T, and B T A T [ 4 4 x + 5y = x y = 7 Compute A T, B T, AB, BA, (AB) T, (BA) T, 4 Verify Property 4 of the transpose of a matrix by showing that if A is an invertible matrix, then so is A T, and (A T ) = (A ) T 5 Suppose A and B are square matrices which commute Prove that (AB) T = A T B T 6 Let A, B, and C be matrices with C invertible If AC = BC, show that A = B 7 Let A be any m n matrix Let x be a vector in R n and let y be a vector in R m Explain the equation y (Ax) = y T Ax
Extra Problems 8 Suppose A and B are invertible matrices and X is a matrix such that AXB = A+B What is X? [ 9 Is in row echelon form? Reduce each of the following matrices to row echelon form In each case, identify the pivots and the pivot columns 4 5 (a) 5 (b) 8 6 4 4 5 8 8 Repeat Exercise for the matrix 5 6 7 7 5 4 4 6 4 4 6 6 4 9 6 Identify the pivot columns, but this time don t go to the trouble of identifying the pivots x Given A = 4 6 6, x = x and b = 4, solve Ax = b expressing your solution as a vector or as a linear combination of vectors as 6 6 x appropriate Write the system x + x + 5x = in the form Ax = b What is A? What 8x + 7x + 9x = 4 is x? What is b? Solve the system, expressing your answer as a vector or a linear combination of vectors as appropriate 4 In solving the system Ax = b, Gaussian elimination on the augmented matrix [A b led to the row echelon matrix [ 5 The variables were x,x,x,x 4 (a) Circle the pivots Identify the free variables (b) Write the solution to the given system as a vector or as a linear combination of vectors, as appropriate 5 Here is a matrix in row echelon form: It represents the final row echelon matrix after Gaussian elimination was applied to the matrix [A b in an attempt to solve Ax = b If possible, find the solution (in vector
Extra Problems form) of Ax = b If there is a solution, state whether this is unique or whether there are infinitely many solutions The vector x is 6 Solve each of the following systems of linear equations by Gaussian elimination and back substitution Write your answers as vectors or as linear combinations of vectors if appropriate (a) x y + z = 4 x + y = x + y 6z = 5 (b) x y + z = 4 (c) x y + z = x + y 6z = 9 x + y 5z = 4 (d) 6x + 8x 5x 5x 4 = 6x + 7x x 8x 4 = 9 8x + x x 9x 4 = 7 Determine whether 6 is a linear combination of the columns of A = 4 8 Determine whether x x is a linear combination of 9 Let v =, v =, v =, and v 4 = and 4 9 5 4 4 7 5 6 9 Determine whether every vector in R is a linear combination of v, v, v, and v 4 Determine whether or not the points (,, ), (,, ), (4, 7, ), (,, 9) are coplanar, that is, lie in a plane If the answer is yes, give an equation of a plane on which the points all lie The planes with equations x + z = 5 and x + y = intersect in a line (a) Why? (b) Find an equation of the line of intersection Suppose we want to find a quadratic polynomial p(x) = a + bx + cx that passes through the points (,), (, ), (5,4)
4 Extra Problems (a) Write the system of equations that must be solved in order to determine a, b, and c (b) Find the desired polynomial Consider the system 5x + y = a 5x 6y = b (a) Under what conditions (on a and b), if any, does this system fail to have a solution? (b) Under what conditions, if any, does the system have a unique solution? Find any unique solution that may exist (c) Under what conditions are there infinitely many solutions? Find these solutions if and when they exist 4 (a) Find a condition on a, b, and c that is both necessary and sufficient for the system to have a solution x y = a 5x + y = b x + y = c (b) Could the system in part (a) have infinitely many solutions? Explain x + x + x = 5 Solve x + x = by Gaussian elimination and back substitution Write x x x = your answer as vectors or as linear combinations of vectors as appropriate 6 (a) Express the zero vector = as a nontrivial linear combination of the columns of A = (b) Use part (a) to show that A is not invertible 7 Find conditions on a, b, and c (if any) such that the system has a unique solution; x + y = y + z = x z = ax + by + cz =
Extra Problems 5 no solution; an infinite number of solutions 8 Write the solution to x + 5x + 7x = 9x = in the form x = x p + x h where x p is a particular solution and x h is a solution of the corresponding homogeneous system 9 Determine whether or not the vectors v =, v =, v =, v 4 = are linearly independent If they are not, write down a nontrivial linear combination of the vectors which equals the zero vector 4 Let E = 4 and let A be an arbitrary n matrix, n (a) Explain the connection between A and EA (b) What is the name for a matrix such as E? (c) Is E invertible? If so, what is its inverse? 4 Suppose A is a n matrix and we would like to modify A by subtracting the second row from the third Write the elementary matrix E for which EA is A modified as specified 4 Let A = 4 5 6 6 8 9 Write an elementary matrix E so that EA = 4 5 6 4 9 4 Let E be the elementary matrix that adds 4 times row one to row two and F the elementary matrix that subtracts times row two from row three (a) Find E, F, EF, and FE without calculation (b) Write the inverses of the matrices in part (a) without calculation 44 What number xwill force an interchange of rows when Gaussian elimination is applied 5 to 4 x? 45 Let A = a a a n be an m n matrix with columns the vectors a,a,,a n and let B be the upper triangular matrix B = b b b n b b n How are the columns b nn
6 Extra Problems of AB related to the columns of B? 46 Write the inverse of each of the following matrices without calculation Explain your reasoning [ (a) A = (b) A = [ 47 Find the inverse of simply by writing it down No calculation is necessary, but explain your reasoning [ 48 Show that the lower triangular matrix L = is invertible by finding a matrix a M with LM = I, the identity matrix 4 49 Let A be a n matrix, E =, D = and P = (a) How are the rows of EA, of DA and of PA related to the rows of A? (b) Find P [ 5 Ian says has no LU factorization but Lynn disagrees Lynn adds the second row to the first, then subtracts the first from the second, like this [ [ [ = U She gets L as the product of certain elementary matrices Who is right? Explain 5 Write L = as the product of elementary matrices (without calculation) 5 and use this factorization to find L 5 Write U = 5 as the product of elementary matrices (without calculation) and use this factorization to find U [ 5 If possible, find an LU factorization of A = and express L as the 7 product of elementary matrices
Extra Problems 7 54 Let A = 4 6 5 and v = 5 (a) Find an elementary matrix E and an upper triangular matrix U such that EA = U (b) Find a lower triangular matrix L such that A = LU (c) Express Av as a linear combination of the columns of A 6 55 Let A = 5 Find an LU factorization of A by moving A to a row 4 echelon matrix (with the elementary row operations), keeping track of the elementary matrices Express L as the product of elementary matrices 56 Find, if possible, an LU factorization of each of the following matrices A: [ 6 5 4 (a) (b) 4 9 6 9 8 [ [ [ 57 Let A = = = LU 6 8 6 5 [ [ x (a) Use this factorization of A to solve the system A = y 9 [ (b) Express as a linear combination of the columns of A 9 [ [ [ 6 58 Use the factorization = to solve Ax = b for x = 5 [ [ [ x 6, with b =, where A = x 4 5 59 Let A = and b = (a) Solve the system Ax = b by applying the method of Gaussian elimination to the augmented matrix (b) Factor A = LU (c) Why is A invertible? (d) Find the first column of A 6 Solve x + x = 8 4x + 9x = as follows:
8 Extra Problems i Write the system in the form Ax = b ii Factor A = LU iii Solve Ly = b for y by forward substitution and Ux = y by back substitution Finally, write the solution to Ax = b in vector form 6 Given A = and b =, i find an LU factorization of A and ii use this to solve Ax = b 6 Find a PLU factorization of A = [ 6 Suppose A and B are symmetric matrices Is AB symmetric (in general)? 64 A matrix A is skew-symmetric if A T = A Show that if A and B are n n skewsymmetric matrices, then so is A + B 65 True or false: The pivots of an invertible matrix A are the nonzero diagonal entries of U in an LU or a PLU factorization 66 If A is a symmetric n n matrix and x is a vector in R n, show that x T A x = Ax [ 67 Determine whether or not A = and B = [ are inverses 7 4 7 68 Find the reduced row echelon form of 5 4 69 Determine whether or not each of the following matrices has an inverse Find the inverse whenever this exists [ (a) (b) 4 6 7 Consider the system of equations x x + x = x + x + x = x + x = i Write this in the form Ax = b ii Solve the system by finding A
Extra Problems 9 iii Express as a linear combination of the vectors,, 7 Given that A is a matrix and 7 Express [ [ A[ 5 as the product of elementary matrices 7 Use Theorem 86 to determine whether or not [ 7 = [ is invertible, find A 74 Given two n n matrices X and Y, how do you determine whether or not Y = X? 75 Is the factorization of an invertible matrix as the product of elementary matrices unique? Explain 76 For each of the following matrices A, i find the matrix M of minors, find the matrix C of cofactors, and compute the products AC T and C T A; ii find deta; iii if A is invertible, find A [ (a) A = 4 7 9 77 Find the determinant of A = row 78 Let A = 4 7 5 (a) Find the matrix of minors (b) Find cofactor matrix C (b) A = 4 4 7 5 expanding by cofactors of the third (c) Find the dot product of the second row of A and the third row of C Could this result have been anticipated? Explain (d) Find the dot product of the first column of A and the second column of C Could this result have been anticipated? Explain (e) Find the dot product of the second column of A and the second column of C What is the significance of this number?
Extra Problems 79 Let A = of A and suppose C = (a) Find the values of c, c and c (b) Find deta c c c (c) Is A invertible? Explain without further calculation (d) Find A, if this exists 8 For what values of x are the given matrices singular? [ x x (a) (b) x + 4 x 7 is the matrix of cofactors 8 Professor G was making up a homework problem for his class when he spilled a cup of coffee over his work (This happens to Prof G all too frequently) All that remained legible was this: A = (a) Find the missing entries (b) Find A 8 Find all values of a which make P(a) = 8 If the matrix of cofactors of a matrix A is ; matrix of cofactors = C = a singular a a + a [ 7 4 84 Given det A = 5 and the matrix of cofactors C =, find A [ 9 4 5, find the matrix A 85 Given that 7 4 5 6 = 8, find
Extra Problems (a) (c) (e) 7 4 5 6 7 4 8 7 4 4 5 6 6 7 5 4 (b) (d) 7 4 5 6 7 4 6 5 86 Let A and B be 5 5 matrices with det( A) = 4 and det B = Find deta, detb and det AB 7 7 9 5 5 87 Let A = 5 77 Find det A, deta, and deta 5 88 Let A = (a) Find the determinant of A expanding by cofactors of the first column (b) Find the determinant of A by reducing it to an upper triangular matrix [ [ 89 Determine whether the vectors, are linearly independent or linearly dependent 4 9 Find the determinants of each of the following matrices by reducing to upper triangular matrices (a) 6 (b) 6 4 4 5 9 9 If P is a permutation matrix and 6 7 PA = 5 5 6 = 7 4 5 6 7 5 7 = LU,
Extra Problems what are the possible values for deta? 9 Give at least three conditions on the rows of a matrix A that guarantee that A is singular 9 Suppose each column of a matrix A is a linear combination of two vectors u and v Find an argument involving the determinant that shows that A is singular 94 Find the following determinants by methods of your choice (a) 6 5 8 4 4 (b) 4 4 (c) 9 6 5 7 6 4 6 8 a b c a g d 95 Suppose d e f = Find b h e g h i c i f 96 Suppose A is a square matrix Explain how an LDU factorization of A could prove 4 helpful in finding the determinant of A and illustrate with A = 4 7 6 6 97 Suppose A is an n n invertible matrix and A = (I A), where I is the n n identity matrix Show that det A is a power of 98 Let A = 5 4 and v = 4 Is v an eigenvector for A? Explain and, if your answer is yes, give the corresponding eigenvalue [ [ [ [ 99 Given A =, determine whether or not each of v 4 =, v =, v =, [ [ v 4 =, v 5 = is an eigenvector of A Justify your answers and state the eigenvalues which correspond to any eigenvectors In each of the following situations, determine whether or not each of the given scalars is an eigenvalue of the given matrix Justify your answers [ 5 7 7 (a) A = ;,,,,5 (b) A = 4 4 ;,,4,5,6 4 4
Extra Problems Let A = [ (a) Show that [ is an eigenvector of A and state the corresponding eigenvalue (b) What is the characteristic polynomial of A? (c) What are the eigenvalues of A? Find the characteristic polynomial, the (real) eigenvalues, and the corresponding eigenspaces of each of the following matrices A [ 5 8 (a) (b) 6 6 4 [ Let A =, let P be the point (x,y) in the Cartesian plane, and let v be the vector OP Let Q be the point such that Av = OQ (a) Find the coordinates of Q (b) Show that the line l with equation y = x is the right bisector of the line PQ joining P to Q; that is, PQ is perpendicular to l and the lines intersect at the midpoint of PQ (c) Describe the action of multiplication of A in geometrical terms (d) Find the eigenvalues and corresponding eigenspaces of A without calculation [ 4 Multiplication by A = is reflection in a line Find the equation of this line Use the fact that the matrix is a reflection to find all eigenvectors and eigenvalues 5 Let λ be an eigenvalue of an n n matrix A and let u,v be eigenvectors corresponding to λ If a and b are scalars, show that A(au + bv) = λ(au + bv) Thus, if au + bv, it is also an eigenvector of A corresponding to λ 6 Suppose the components of each row of an n n matrix A add to 7 Prove that 7 is an eigenvalue of A 7 Suppose A, B, and P are n n matrices with P invertible and PA = BP Show that every eigenvalue of A is an eigenvalue of B 8 What are the eigenvalues of? What are the corresponding eigenvalues?
4 Extra Problems 9 Let v be an eigenvector of a matrix A with corresponding eigenvalue λ Is v an eigenvector of 5A? If so, what is the corresponding eigenvalue? Explain (a) Suppose a matrix A is similar to the identity matrix What can you say about A? Explain [ (b) Can be similar to the identity matrix? Explain [ (c) Is A = diagonalizable? Explain Suppose A, B, and C are n n matrices If A is similar to B and B is similar to C, show that A is similar to C [ [ Use the given fact that A = and B = are similar matrices to 4 8 find the determinant of B and the characteristic polynomial of B [ (a) Find all eigenvalues and eigenvectors of A = (b) Explain why the results of part (a) show that A is not diagonalizable [ 4 Let A = (a) Show that λ = is an eigenvalue of A and find the corresponding eigenspace (b) Given that the eigenvalues of A are and, what is the characteristic polynomial of A? (c) Why is A diagonalizable? [ (d) Given that is an eigenvector of A corresponding to λ =, find a matrix P [ such that P AP = [ 6 (e) Let Q = Find Q 4 AQ without calculation 5 Let A = 4 5 5 5 4 (a) What is the characteristic polynomial of A? (b) Why is A diagonalizable? (c) Find a matrix P such that P AP = 4
Extra Problems 5 6 Determine whether or not each of the matrices A given below is diagonalizable If it is, find an invertible matrix P and a diagonal matrix D such that P AP = D If it isn t, explain why (a) A = 7 Let A = [ [ (b) A = (c) A = (a) Find an invertible matrix P and a diagonal matrix D such that P AP = D (b) Find a matrix B such that B = A [Hint: First, find a matrix D such that D = D 8 One of the most important and useful theorems in linear algebra says that a real symmetric matrix A is always diagonalizable in fact, orthogonally diagonalizable; that is, there is a matrix P with orthogonal columns such that P AP is a diagonal matrix) Illustrate this fact with respect to the matrix A = [ 9 Repeat Exercise 8 for A = 5 5 [ 96 96 47 Prove Theorem in general; that is, if x,x,,x k are eigenvectors of a matrix A whose corresponding eigenvalues λ,λ,,λ k are all different, then x,x,,x k are linearly independent [Hint: If the result is false, then there are scalars c,c,,c k, not all, such that c x + c x + + c k x k = Omitting the terms with c i = and renaming the terms that remain, we may assume that c x + c x + + c l x l = ( ) for some l, where c, c,, c l Among all such equations, suppose the one exhibited is shortest; that is, whenever we have an equation like ( ) (and no coefficients ), there are at least l terms Note that l >, since otherwise c x = and x gives c =, a contradiction Now imitate the proof of Theorem Given that a matrix A is similar to the matrix, what s the characteristic polynomial of A? Explain 4 5 If A is a diagonalizable matrix with all eigenvalues nonnegative, show that A has a square root; that is, A = B for some matrix B [Hint: Exercise 7