MÄLARDALEN UNIVERSIY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson EXAMINAION IN MAHEMAICS MAA15 Linear Algebra Date: 017-06-09 Write time: 5 hours Aid: Writing materials, ruler his examination consists of eight randomly ordered problems each of which is worth at maximum 5 points. he maximum sum of points is thus 40. he pass-marks, 4 and 5 require a minimum of 18, 6 and 4 points respectively. he minimum points for the ECS-marks E, D, C, B and A are 18, 0, 6, and 8 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. Let q be a quadratic form on E defined by q(u) = x + y z + xz, where x, y, z are the coordinates of u relative to the standard basis. In which (closed) interval is q(u) when u = and for which vectors are the minimum and the maximum respectively attained?. Let the linear space which is spanned by the functions p 1, p, p, where p k (x) = x k, be equipped with the inner product p q = 1 p(x)q(x) dx. Find an ON-basis 1 for the linear space. x 1 = 7 x 1 + x 6 x. Prove that the relationships x = x 1 x + x defines a change-of-basis x = x 1 + x x between two ordered bases e 1, e, e and ẽ 1, ẽ, ẽ, where x 1, x, x and x 1, x, x are the coordinates of a vector u with respect to respectively of the bases. Also, find the coordinates of the vector 5e 1 e +e with respect to the basis ẽ 1, ẽ, ẽ. 4. Find, relative to the standard basis, the matrix of the linear operator F : R R whose kernel is equal to span{( 0, 1)}, and where span{(, 0), ( 4, 1)} is the eigenspace of F corresponding to the eigenvalue. 5. Find the length of the orthogonal projection of the vector e 1 4e + e on the vector e 1 +e e in the Euclidean space E for which the inner product is fixed as u v = 4x 1 y 1 +5x y +11x y +4(x 1 y +x y 1 )+7(x y +x y )+6(x y 1 +x 1 y ), where x 1, x, x and y 1, y, y are the coordinates of u and v respectively relative to the basis e 1, e, e. 6. he linear operator F : R R has relative to the standard basis the matrix 1 0 0 1 β β where β R. Find the numbers β for which the operator är diagonalizable, and state a basis of eigenvectors for each of these β. 7. he linear transformation F : R 5 R 4 has the matrix 6 4 16 5 9 8 4 7 11 4 11 1 relative to the standard bases for R 5 and R 4. Find a basis for the kernel of F and a basis for the image of F. {( ) ( ) ( ) ( )} 8. Find a basis for the subspace span 6 5, 8 1 5 7, 1 4 1, 8 16 5 10 of the vector space of all real-valued -matrices. Also, find out whether the matrix belong to the subspace or not. ( 1 9 ) 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 ENAMEN I MAEMAIK MAA15 Linjär algebra Datum: 017-06-09 Skrivtid: 5 timmar Hjälpmedel: Skrivdon, linjal Denna tentamen består av åtta stycken om varannat slumpmässigt ordnade uppgifter som vardera kan ge maximalt 5 poäng. Den maximalt möjliga poängsumman är således 40. För godkänd-betygen, 4 och 5 krävs minst 18, 6 respektive 4 poäng. För ECS-betygen E, D, C, B och A krävs 18, 0, 6, respektive 8 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. Låt q vara en kvadratisk form på E definierad enligt q(u) = x + y z + xz där x, y, z är koordinaterna för u relativt standardbasen. I vilket (slutet) intervall ligger q(u) då u = och för vilka vektorer antas minimum respektive maximum?. Låt det linjära rum som spänns upp av funktionerna p 1, p, p, där p k (x) = x k, vara utrustat med skalärprodukten p q = 1 p(x)q(x) dx. Bestäm en ON-bas 1 för det linjära rummet. x 1 = 7 x 1 + x 6 x. Bevisa att sambanden x = x 1 x + x definierar ett basbyte mellan x = x 1 + x x två ordnade baser e 1, e, e och ẽ 1, ẽ, ẽ, där x 1, x, x och x 1, x, x är koordinaterna för en vektor u med avseende på respektive av baserna. Bestäm även koordinaterna för vektorn 5e 1 e + e med avseende på basen ẽ 1, ẽ, ẽ. 4. Bestäm, relativt standardbasen, matrisen för den linjära operator F : R R vars nollrum är lika med span{( 0, 1)}, och där span{(, 0), ( 4, 1)} är egenrummet till F motsvarande egenvärdet. 5. Bestäm längden av den ortogonala projektionen av vektorn e 1 4e +e på vektorn e 1 + e e i det euklidiska rum E för vilket skalärprodukten är fixerad till u v = 4x 1 y 1 +5x y +11x y +4(x 1 y +x y 1 )+7(x y +x y )+6(x y 1 +x 1 y ), där x 1, x, x och y 1, y, y är koordinaterna för u respektive v relativt basen e 1, e, e. 6. Den linjära operatorn F : R R har relativt standardbasen matrisen 1 0 0 1 β β där β R. Bestäm de tal β för vilka operatorn är diagonaliserbar, och ange en bas av egenvektorer till F för var och en av dessa β. 7. Den linjära avbildningen F : R 5 R 4 har matrisen 6 4 16 5 9 8 4 7 11 4 11 1 relativt standardbaserna för R 5 och R 4. Bestäm en bas för F :s nollrum och en bas för F :s värderum. {( ) ( ) ( ) ( )} 8. Bestäm en bas för underrummet span 6 5, 8 1 5 7, 1 4 1, 8 16 5 10 till det linjära) rummet av alla reellvärda -matriser. Utred även huruvida matrisen tillhör underrummet eller inte. ( 1 9 If you prefer the problems formulated in English, please turn the page.
MÄLARDALEN UNIVERSIY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson Examination 017-06-09 1. he quadratic form q (u) is in the interval [, ]. he minimum is attained for ± 1 ( 0, 1) and the maximum for ± (0,0). An ON-basis for the linear space is e.g. p 5 p, 1 7 ( 5 p p1 ). Proof: he matrix S in the matrix ~ version X = SX of the given relationships is invertible and thus S is a change-of-basis matrix in changing from the basis e 1, e, to e ~ ~,. he coordinates of 5e 1 e + relative to the basis e ~ ~, are,7,5, i.e. 5e e + e = e ~ + 7 e ~ + 5 e ~ 1 1 EXAMINAION IN MAHEMAICS MAA15 Linear algebra EVALUAION PRINCIPLES with POIN RANGES Academic year: 016/17 Maximum points for subparts of the problems in the final examination 1p: Correctly found the eigenvalues of the symmetric operator corresponding to the quadratic form (QF) 1p: Correctly identified the diagonal form of the QF 1p: Correctly, from the diagonalized QF, identified the interval in which q (u) lies as u = 1 1p: Correctly found the vectors for which q attains minimum as u = 1 1p: Correctly found the vectors for which q attains maximum as u = 1 1p: Correctly normalized one of the three functions, e.g. p into e 1 1p: Correctly found and explicitly evaluated a nonzero function f which is orthogonal to the function e e.g. the function p e1 e1 p 1p: Correctly normalized the function f into e 1p: Correctly found and explicitly evaluated a nonzero function f which is orthogonal to the two functions e 1 and e, e.g. the function p e1 e1 p e e p 1p: Correctly normalized the function f into e, and correctly summarized e 1, e, as an ON-basis for the given linear space p: Correctly proved why the relationships define a changeof-basis matrix S between two ordered bases e 1, e, and e ~ ~, e 1p: Correctly found that the coordinates of 5e 1 e + with respect to the basis e ~ ~, are given by the coordinate matrix S 1 X, where X is equal to the coordinate matrix ( 5 1 ) 1 1p: Correctly found the matrix S 1p: Correctly found the coordinates of 5e 1 e + with respect to the basis e ~ ~, e 4. F has the matrix 9 9 1 0 10 0 5 1 1 relative to the standard basis 1p: Correctly interpreted the vector spanning the kernel, i.e. that A ( 1 0 1) = (0 0 0) for the matrix A of F relative the standard basis 1p: Correctly interpreted the vectors spanning the eigenspace, i.e. that A ( 1 0) = ( 1 0) and A ( 1 4 1) = (1 4 1) for the matrix A of F 1p: Correctly, based on the three given conditions, formulated an equation for the matrix A p: Correctly solved the equation for the matrix A 1 ()
5. ( e 4e + e ) 1 e + e e = 1 5 1p: Correctly stated the (formula) expression for the orthogonal projection of e 1 4e + on e1 + e 1p: Correctly interpreted how the given inner product is applied 1p: Correctly found the inner product of the vectors e 1 4e + and e1 + e 1p: Correctly found the length of the vector e1 + e 1p: Correctly found the length of the orthogonal projection 6. he linear operator F is diagonalizable iff β 4, and a basis of eigenvectors is e.g. ( 1), ( 0,0,1), ( 4 β,8 β,1 + β ) 7. A basis for the image of F is e.g. ( 6,,,4), ( 4,5, 4,). A basis for the kernel of F is e.g. (, 0,0), ( 0,0), (,0,0,1) 1p: Correctly found that the linear operator (LO) is diagonalizable if β 4 1p: Correctly found that the LO is diagonalizable if β = 1 1p: Correctly found that the LO is not diagonalizable if β = 4 1p: Correctly for β = 1 found a basis of eigenvectors 1p: Correctly for β 4 found a basis of eigenvectors p: Correctly found a basis for the image of F p: Correctly found a basis for the kernel of F 8. A basis for the given subspace of M (, R ) is e.g. 6 5 he matrix the subspace, 8 1, 1 4. 5 7 1 1 9 does not belong to p: Correctly initiated an analysis of the relation between the five vectors in M (, R ), and correctly found the rowechelon form of the augmented coordinate matrix of the vectors spanning the subspace of M (, R ) together with the fifth vector p: Correctly from the row-echelon form identified a basis for the subspace of M (, R ) 1p: Correctly from the row-echelon form concluded that the fifth vector does not belong to the subspace of M (, R ) ()