MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson EXAMINATION IN MATHEMATICS MMA19 Linear Algebra Date: 015-08-1 Write time: 5 hours Aid: Writing materials This examination consists of eight randomly ordered problems each of which is worth at maximum 5 points. The maximum sum of points is thus 40. The pass-marks 3, 4 and 5 require a minimum of 18, 6 and 34 points respectively. Solutions are supposed to include rigorous justifications and clear answers. All sheets with solutions must be sorted in the order the problems are given in. Especially, avoid to write on back pages of solution sheets. 1. In the standard bases for R 4 and R, find the matrix of a linear transformation F : R 4 R whose image and kernel are equal to span{(, 1), (3, )} and span{(3, 1,, 1), (1,, 1, 1)} respectively.. Let (x, y, z) denote the coordinates of a point in an orthonormal system. Prove that the equation + (x + y) = 4(x + y)z describes a cylindrical surface and determine the specific type. Also, find (expressed in the given basis) the coordinates of a vector parallel with a cylindrical axis of the surface. 3. Assume that p k, k = 0, 1,, are the real-valued polynomial functions which have the polynomials 1, x and x respectively. For which real numbers a is the dimension of the linear span P of the three polynomial functions p 0 3p 1 + p, p 0 + ap 1 and 4p 1 ap less than 3? Also, determine for these a whether the polynomial function p 0 + 4p 1 + ap belongs to P or not. 4. Let e 1, e, e 3 be a basis for the linear space L, and introduce the vectors ẽ 1, ẽ, ẽ 3 according to ẽ 1 + 3ẽ ẽ 3 = 7e 1 + 9e + 4e 3, ẽ 1 = 3e 1 e e 3, ẽ ẽ 3 = e 1 + 5e + e 3. Verify that also ẽ 1, ẽ, ẽ 3 is a basis for L, and determine the coordinates of the vector ẽ 1 4ẽ + 3ẽ 3 in the basis e 1, e, e 3. 5. The linear operator F : R 3 R 3 has relative to the standard basis the matrix 0 0 1 β β 1 1 1 where β R. Find the β for which the operator är diagonalizable, and state a basis of eigenvectors for each of these β. 6. In the linear space R 3, the scalar product, given by 3 (x 1, x, x 3 ), (y 1, y, y 3 ) = a ij x i y j is introduced. Determine the coefficients a ij in the expression for, knowing that the vectors (1, 0, 1), (1, 1, 1), (0, 1, 1) constitute an ON-basis. 7. Let L = span{(, 1, 1, 1), (1, 1, 1, 1)} be equipped with the standard scalar product, i.e. L E 4. Find an orthogonal basis for the orthogonal complement L of L, where L = {u E 4 : u v = 0 for all v L}. i,j=1 8. The linear operator F projects each vector in E 3 orthogonally on M = {(x 1, x, x 3 ) E 3 : x 1 + x + x 3 = 0}. Find the matrix of F relative to the standard basis. Om du föredrar uppgifterna formulerade på svenska, var god vänd på bladet.
MÄLARDALENS HÖGSKOLA Akademin för utbildning, kultur och kommunikation Avdelningen för tillämpad matematik Examinator: Lars-Göran Larsson TENTAMEN I MATEMATIK MMA19 Linjär algebra Datum: 015-08-1 Skrivtid: 5 timmar Hjälpmedel: Skrivdon Denna tentamen består av åtta om varannat slumpmässigt ordnade uppgifter som vardera kan ge maximalt 5 poäng. Den maximalt möjliga poängsumman är således 40. För betygen 3, 4 och 5 krävs minst 18, 6 respektive 34 poäng. Lösningar förutsätts innefatta ordentliga motiveringar och tydliga svar. Samtliga lösningsblad skall vid inlämning vara sorterade i den ordning som uppgifterna är givna i. Undvik speciellt att skriva på baksidor av lösningsblad. 1. Bestäm i standardbaserna för R 4 och R avbildningsmatrisen för en linjär avbildning F : R 4 R vars värderum och nollrum är lika med de linjära höljena [(, 1), (3, )] respektive [(3, 1,, 1), (1,, 1, 1)].. Låt (x, y, z) beteckna en punkts koordinater i ett ON-system. Visa att ekvationen + (x + y) = 4(x + y)z beskriver en cylindrisk yta och bestäm den specifika typen. Bestäm även (uttryckt i den givna basen) koordinaterna för en vektor som är parallell med en cylinderaxel till ytan. 3. Antag att p k, k = 0, 1,, är de reellvärda polynomfunktioner som har polynomen 1, x respektive x. För vilka reella tal a är dimensionen av det linjära höljet P av de tre polynomfunktionerna p 0 3p 1 + p, p 0 + ap 1 och 4p 1 ap mindre än 3? Avgör även för dessa a om polynomfunktionen p 0 + 4p 1 + ap tillhör P eller ej. 4. Låt e 1, e, e 3 vara en bas för det linjära rummet L, och introducera vektorerna ẽ 1, ẽ, ẽ 3 enligt ẽ 1 + 3ẽ ẽ 3 = 7e 1 + 9e + 4e 3, ẽ 1 = 3e 1 e e 3, ẽ ẽ 3 = e 1 + 5e + e 3. Visa att ẽ 1, ẽ, ẽ 3 också är en bas för L, och bestäm koordinaterna för vektorn ẽ 1 4ẽ + 3ẽ 3 basen e 1, e, e 3. 5. Den linjära operatorn F : R 3 R 3 har i standardbasen matrisen 0 0 1 β β 1 1 1 där β R. Bestäm de β för vilka operatorn är diagonaliserbar, och ange en bas av egenvektorer till F för var och en av dessa β. 6. I det linjära rummet R 3 är skalärprodukten, enligt 3 (x 1, x, x 3 ), (y 1, y, y 3 ) = a ij x i y j införd. Bestäm koefficienterna a ij i uttrycket för, utifrån vetskapen om att vektorerna (1, 0, 1), (1, 1, 1), (0, 1, 1) utgör en ON-bas. 7. Låt L vara det linjära höljet [(, 1, 1, 1), (1, 1, 1, 1)] utrustat med standardskalärprodukten, dvs L E 4. Bestäm en ON-bas för det ortogonala komplementet L till L, där L = {u E 4 : u v = 0 för alla v L}. i,j=1 8. Den linjära operatorn F projicerar varje vektor i E 3 ortogonalt på M = {(x 1, x, x 3 ) E 3 : x 1 + x + x 3 = 0}. Bestäm avbildningsmatrisen för F i standardbasen. If you prefer the problems formulated in English, please turn the page.
MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson Examination 015-08-1 1. F could e.g. have the matrix 3 5 13 1 8 11. Part 1: A proof Part : Hyperbolic cylinder Part 3: The coordinates of a vector parallel with a cylindrical axis of the surface is e.g. ( 1, 1,0) EXAMINATION IN MATHEMATICS MMA19 Linear algebra EVALUATION PRINCIPLES with POINT RANGES Academic year: 014/15 Maximum points for subparts of the problems in the final examination p: Correctly (with an explanation) found that two of the columns of the matrix should consist of the coordinates of two linearly independent linear combinations of the vectors which span the image 1p: Correctly interpreted the kernel as spanned by those column vectors which are mapped on the zero vector, i.e. T that the matrix relations A ( 3 1 1) = 0 and T A ( 1 1 1) = 0 are valid p: Correctly determined the third and the forth columns of the matrix 1p: Correctly proven that + ( x + y) = 4( x + y) z describes a cylindrical surface p: Correctly determined the type of cylindrical surface p: Correctly found the coordinates of a vector parallel with a cylindrical axis of the surface 3., if ( a = 8) ( a = ) dim( P ) = 3, if a 8, p + p + ap P if a 0 4 1 1p: Correctly initiated an analysis of the relation between the four functions in P, and correctly found the reduced echelon form of the augmented coordinate matrix of the functions in the span for P together with the fourth function p: Correctly from the reduced echelon form identified a 4 basis for the subspace M of R p: Correctly from the reduced echelon form identified the value of a for which M contains the fourth function 4. The vectors e 1, e, e3 constitute a basis since the matrices A and B in the relation e A = eb are invertible and make up a change-of-basis matrix 1 S = BA. koord e,, ( e1 4e + 3e3 ) = (1, 9, 3) 1 e e3 3p: Correctly found that e 1, e, e3 is a basis, where 1p is given for having correctly realized that not only the matrix A but also the matrix B in the relation e A = eb must be invertible, and (1+1)p is given for proofs of A and B being invertible 1p: Correctly found that the coordinates of the vector e 1 4 e + 3e3 in the basis e 1, e, e3 are given by the coordinate matrix S X, where X is equal to the T coordinate matrix ( 1 4 3) 1p: Correctly evaluated the expressions for the coordinates of the vector e 4 e + 3 e3 in the basis e 1, e, e3 1 1 ()
5. The linear operator is diagonalizable iff β 0, A basis of eigenvectors is e.g. ( 1, 1,0), ( 1,1,), ( 1,( β 1), β ) 6. The coefficients a ij are equal to the matrix elements of the matrix 3 3 6 4 4 3 1p: Correctly for β = 0 found that the linear operator is not diagonalizable 1p: Correctly for β = found that the linear operator is not diagonalizable 1p: Correctly for β 0, found that the linear operator is diagonalizable p: Correctly for β 0, found a basis of eigenvectors -------------------------------- One scenario ------------------------------------------ 1p: Correctly noted that the matrix of the scalar product relative to the given ON-basis ( 1,0, 1), ( 1,1,1), ( 0,1,1) is equal to the identity matrix, and that the matrix relative to the standard basis has the matrix elements a ij p: Correctly noted that the matrix relative to the standard T 1 basis is equal to the matrix ( SS ), where S is the change-of-basis matrix from the standard basis to the ONbasis T 1 p: Correctly found the matrix ( SS ) ----------------------------- Another scenario --------------------------------------- p: Correctly stated the orthogonality and norm conditions for the ON-basis ( 1,0, 1), ( 1,1,1), ( 0,1,1), i.e. 1 = (1,0, 1) (1,0, 1) = a 11 a13 + a33, 0 = (1,0, 1) (1,1,1) = a11 + a1 a3 a33 etc, where a ji = aij since an inner product for vector spaces over real numbers is symmetric in its arguments 3p: Correctly solved the system of equations, where the number of unknowns are 9 3 = 6 since the scalar product is symmetric (implying that a = a ) ji ij 7. E.g. (, 1,3,0), ( 6,3,5,14) 1p: Correctly noted that the orthogonal complement of L may be stated as 4 { x E x + x x + x = 0, x x x + x = 0 } : 1 3 4 1 3 4 p: Correctly found a two-dimensional basis for p: Correctly orthogonalized the basis for L L 8. 1 1 3 1 1 1 1 1 1p: Correctly from the condition for M identified two linearly independent vectors which span M and one vector which spans the orthogonal complement M of M 1p: Correctly noted that F maps vectors in M on themselves, i.e. F ( u) = u for each u M 1p: Correctly noted that F maps vectors in M on the zero vector, i.e. F ( v) = 0 for each v M 1 1p: Correctly on the form CB, and in the standard basis, found the matrix A of the linear transformation F 1p: Correctly found the explicit expression for the matrix A ()