, m 3 = 3. Determine for each real α and for each real β 0 the geometric meaning of the equation x 2 + 2y 2 + αz 2 + 2xz 4yz = β.

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1 MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson EXAMINATION IN MATHEMATICS MMA9 Linear Algebra Date: 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, 4 and 5 require a minimum of 8, 6 and 4 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.. The linear operator F : E E has the matrix 4 A = 0. 4 relative to the standard basis. Prove that F is diagonalizable, i.e. that there exists a basis of eigenvectors of F, and determine the matrix à of F relative to that basis. Finally, find an orthogonal change-of-basis matrix S in the relation à = S AS between the matrices of F in the two bases.. Let m = ( ), m = ( ), m = ( Prove that m, m, m is a basis for the linear space M of all real-valued - matrices with the trace (the sum of the ( diagonal) elements) equal to zero. Also, determine the coordinates of the matrix 9 in the basis m, m, m.. Determine for each real α and for each real β 0 the geometric meaning of the equation x + y + αz + xz 4yz = β. 4. Find an orthonormal basis for the linear space 6 9 {(x, x, x, x 4 ) E 4 : x + x + x + x 4 = 0}. 5. Let P be the linear space of polynomial functions p of degree, and define by F (p)(x) = p(x + ) a linear operator F : P P. Find the matrix of F in the basis p p 0, p, p 0 + p, where p n (x) = x n. 6. For each value of a, find the dimension of, and a basis for, the linear span span{(a +, 0, a +, 0), (,, 0, 6), (,, a, 4), (a +,,, a )} R The linear transformation F : R 5 R 4 has in the standard bases the matrix Find the kernel and the image of F, and express them as linear spans generated by a set of basis vectors for each space respectively. Also, specify explicitly the dimensions of the two spaces. 8. Determine a unit vector n that is orthogonal to both e +e +e and e +e +e in the inner product space E for which the scalar product is fixed as u v = 6x y + x y + x y (x y + x y ) + (x y + x y ) 4(x y + x y ), where (x, x, x ) and (y, y, y ) are the coordinates of u and v respectively in the basis e, e, e. ). Om du föredrar uppgifterna formulerade på svenska, var god vänd på bladet.

2 MÄLARDALENS HÖGSKOLA Akademin för utbildning, kultur och kommunikation Avdelningen för tillämpad matematik Examinator: Lars-Göran Larsson TENTAMEN I MATEMATIK MMA9 Linjär algebra Datum: 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, 4 och 5 krävs minst 8, 6 respektive 4 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.. Den linjära operatorn F : E E ges i standardbasen av matrisen 4 A = Låt Visa att F är diagonaliserbar, dvs att det finns en bas av egenvektorer till F, och bestäm F :s matris à i denna bas. Bestäm slutligen en ortogonal basbytesmatris S i sambandet à = S AS mellan avbildningsmatriserna i de två baserna. m = ( ), m = ( ), m = ( Visa att m, m, m är en bas för det linjära rummet M av alla reellvärda - matriser med spåret (summan ( av diagonalelementen) ) lika med noll. Bestäm även koordinaterna för matrisen 9 i basen m, m, m Bestäm för varje reellt α och för varje reellt β 0 den geometriska innebörden av ekvationen x + y + αz + xz 4yz = β. 4. Bestäm en ON-bas för det linjära rummet {(x, x, x, x 4 ) E 4 : x + x + x + x 4 = 0}. 5. Låt P vara det linjära rummet av polynomfunktioner p av grad, och definiera genom F (p)(x) = p(x + ) en linjär operator F : P P. Bestäm avbildningsmatrisen för F i basen p p 0, p, p 0 + p, där p n (x) = x n. 6. Bestäm för varje värde på a dimensionen av, och en bas för, det linjära höljet [(a +, 0, a +, 0), (,, 0, 6), (,, a, 4), (a +,,, a )] R Den linjära avbildningen F : R 5 R 4 har i berörda standardbaser matrisen Bestäm F :s nollrum och värderum, och uttryck dem som linjära höljen genererade av en uppsättning basvektorer för respektive rum. Ange även explicit dimensionerna av de två rummen. ). 8. Bestäm en enhetsvektor n som är ortogonal mot såväl e +e +e som e +e +e i det euklidiska rum E för vilket skalärprodukten är fixerad som u v = 6x y + x y + x y (x y + x y ) + (x y + x y ) 4(x y + x y ), där (x, x, x ) och (y, y, y ) är koordinaterna för u respektive v i basen e, e, e. If you prefer the problems formulated in English, please turn the page.

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8 MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson EXAMINATION IN MATHEMATICS MMA9 Linear algebra EVALUATION PRINCIPLES with POINT RANGES Academic year: 04/5 Examination A basis of eigenvectors e, e, e of the linear operator F is e.g. e = ( e e ) e = ( e 4e + e) e = ( + + ) e 6 e e 0 0 A = 0 0, 4 S = Maximum points for subparts of the problems in the final examination p: Correctly proven that F is diagonalizable, by e.g. determining three linearly independent eigenvectors (p for each fully determined eigenspace) p: Correctly determined the matrix A of F relative the basis of eigenvectors e, e, e p: Correctly orthogonalized the column vectors of an orthogonal change-of-basis matrix S p: Correctly normalized the column vectors of an orthogonal change-of-basis matrix S. Proof The coordinates of m, m, m are ( 5,, 4 ). If β = 0 then α < : an elliptic cone α = : a line α > : a point (the origin) in the basis If β > 0 and α < : a one-sheeted hyperboloid α = : an elliptic cylinder α > an ellipsoid 4. An orthonormal basis is e.g. (, 0, 0, ), (,, 0, ) 6 (,,,, ) p: Correctly proven that the vectors m, m, m is a basis for the linear space M p: Correctly determined the coordinates of the matrix 9 in the basis m, m, m 6 9 p: Correctly completed the squares of the quadratic form h α (u) so that conclusions about the meaning of the equation h α (u) = β for different values of α and β can be drawn correctly p: Correctly determined the geometric meaning in the case ( β = 0) ( α < ) p: Correctly determined the geometric meanings in the cases ( β = 0) ( α = ) and ( β = 0) ( α > ) respectively p: Correctly determined the geometric meaning in the case ( β > 0) ( α = ) p: Correctly determined the geometric meaning in the cases ( β > 0) ( α < ) and ( β > 0) ( α > ) respectively p: Correctly found that the linear space is spanned by e.g. the vectors (, 0, 0, ), ( 0,, 0, ) and ( 0, 0,, ), here denoted by u, u, u respectively p: Correctly initiated a Gram-Schmidt process to find an orthonormal basis e, e, e, and correctly as a first step in the process normed u to e p: Correctly performed a second step in the G-S process by defining a vector f = u u e e (which a) is in the linear space, b) is not equal to the zero vector, and c) is orthogonal to e ), by evaluating u u e e, and by norming f to e p: Correctly performed a third step in the G-S process by defining a vector f = u u e e u e e (which a) is in the linear space, b) is not equal to the zero vector, and c) is orthogonal to e and e ), by evaluating u u e e u e e, and by norming f to e ()

9 Another scenario p: Correctly determined that F ( p0) = p0, F ( p) = p0+ p and F ( p) = 4 p0+ 4 p+ p p: Correctly formulated the matrix A of F relative the basis p 0, p, p p: Correctly determined the change-ofbasis matrix S for a change from the basis p 0, p, p to basis e, e, e p: Correctly determined the inverse of the matrix S p: Correctly found and evaluated the matrix product S AS as the matrix of F relative the basis e, e, e One scenario p: Correctly determined that F maps the st of the three polynomial base functions to the second, i.e. that F ( e) = e where e= p p0 and e = p p: Correctly determined that F maps the nd of the three polynomial base functions to the linear combination p + p 0, and that the latter can be reformulated as p ( p p0) by which follows that F( e) = e e p: Correctly determined that F maps the rd of the three polynomial base functions to the linear combination p + 4 p + 5p0, and that the latter can be reformulated as ( p0 + p) + 6 p ( p p0) by which follows that F( e) = e+ 6e e p: Correctly formulated the matrix of F relative to the basis e, e e, 6. if ( a = ) ( a = ) dim LS ( a) = 4 if a, a = : A basis for the LS is e.g. (,,0,),(,,,4),(,,, 4) a =: A basis for the LS is e.g. (,0,,0),(,,0,),(4,,, ) a,: A basis for the LS is e.g. the four vectors in the definition of the span (but of course also e.g. the 4 standard basis for R ) p: Correctly initiated an analysis of the vectors defining the linear span, correctly determined the reduced rowechelon form of the coordinate matrix of the vectors, and correctly deduced that the special cases (dimension of the linear span less than 4) occur if a = or a = p: Correctly in the case a = found the dimension and a basis for the linear span (LS) p: Correctly in the case a = found the dimension and a basis for the linear span (LS) p: Correctly in the case a, found the dimension and a basis for the linear span (LS) 7. ker(f) = span{ (,,,0,0),(,,0,,0) } im(f) = span{ (,,, 4),(,,,),(,,, ) } dim(ker (F)) = dim(im (F)) = 8. The vectors ( e + 4e e ) 6 ( e 4e + e 6 are the two possibilities for n ) p: Correctly determined the reduced row-echelon form of the matrix p: Correctly determined the dimensions of ker(f) and im(f ) respectively p: Correctly determined a linear span of two vectors for the kernel of F p: Correctly determined a linear span of a three vectors for the image of F p: Correctly formulated and evaluated the condition that n = a e + be + ce is orthogonal to e + e + e p: Correctly formulated and evaluated the condition that n = a e + be + ce is orthogonal to e + e + e p: Correctly solved the system of linear equations for finding two (in terms of the third) of the coordinates ( a, b, c) of the vector n in the basis e, e, e p: Correctly evaluated the condition for n being normed ()