UME A UNIVERSITY 2012-02-07 9:00{13:00 · (b) Ber akna avst andet mellan planen. (2 p) Information r orande denna dugga: En engelsk version av duggan ar tillg anglig p a andra sidan

UMEA UNIVERSITYDepartment of Mathematicsand Mathematical StatisticsGerold Jager

Pre-exam in mathematicsLinear algebra

2012-02-079:00–13:00

1. Compute the following matrix: (2 p)

3 ·

1 −23 22 7

· ( 4 35 −2

)T

2. Compute the determinant of the following matrix: (2 p)

1 0 1 01 1 0 03 2 3 12 0 0 −1

3. Solve the following equation:

X ·(

1 32 5

)=

(8 −23 4

)In other words, find a 2× 2 matrix X such that this equation is true. (2 p)

4. Solve the following linear system: (2 p)

−x1 + 2x2 − x3 = 32x1 − 3x2 + 2x3 = −1−x1 + 2x2 = −2

5. Let u = (4,−3, 1) and a = (2, 3− 1).

Compute the orthogonal projection of u on a, i.e., compute w1 = proja(u). (2 p)

6. Let u = (2,−3, 1), v = (4, 2− 1), and w = (1, 0− 6) be vectors with the same initial point.

Do these vectors lie in the same plane? Explain your answer. (2 p)

7. Let u = (1,−4, 2), v = (2, 0,−2) be vectors.

(a) Find a vector w ∈ R3 which is orthogonal to u and v. (1 p)

(b) Find a vector with norm 1 which is orthogonal to v. (1 p)

8. Let u = (−1,−3, 1), v = (1, 4, 3) and w = (5,−1, 2).

Compute the volume of the parallelepiped determined by these three vectors. (2 p)

Information regarding this pre-exam:

• A Swedish version of the pre-exam is available on the opposite site of this sheet. Solutionsmay be written in Swedish or English.

• In each assignment write each intermediate step leading to the final result. Solutions withoutthese intermediate steps will not get any points, even if they are correct.

• The number of bonus points for the exam is the number of points reached in this pre-examdivided by 4.

• Only non-symbolic calculators are allowed.

Good luck!

1

UMEA UNIVERSITETInstitutionen for matematikoch matematisk statistikGerold Jager

Dugga i matematikLinjar algebra

2012-02-079:00–13:00

1. Berakna foljande matris: (2 p)

3 ·

1 −23 22 7

· ( 4 35 −2

)T

2. Berakna determinanten till foljande matris: (2 p)

1 0 1 01 1 0 03 2 3 12 0 0 −1

3. Los foljande ekvation:

X ·(

1 32 5

)=

(8 −23 4

)Med andra ord, hitta en 2× 2-matris X sadan att likheten ar sann. (2 p)

4. Los foljande linjara system: (2 p)

−x1 + 2x2 − x3 = 32x1 − 3x2 + 2x3 = −1−x1 + 2x2 = −2

5. Lat u = (4,−3, 1) och a = (2, 3,−1).

Berakna den ortogonala projektionen av u pa a, dvs. berakna w1 = proja(u). (2 p)

6. Lat u = (2,−3, 1), v = (4, 2,−1), och w = (1, 0,−6) vara vektorer med samma utgangspunkt.

Ligger dessa vektorer i samma plan? Motivera ditt svar. (2 p)

7. Lat u = (1,−4, 2), v = (2, 0,−2) vara vektorer.

(a) Hitta en vektor w ∈ R3 som ar ortogonal mot u och v. (1 p)

(b) Hitta en vektor med norm 1 som ar ortogonal mot v. (1 p)

8. Lat u = (−1,−3, 1), v = (1, 4, 3) och w = (5,−1, 2).

Berakna volymen av den parallellepiped som bestams av dessa tre vektorer. (2 p)

Information rorande denna dugga:

• En engelsk version av duggan ar tillganglig pa andra sidan av detta blad. Losningar kanskrivas pa svenska eller engelska.

• I varje uppgift, skriv varje mellanled som leder fram till ditt svar. Losningar utan dessamellanled kommer ej att ges nagra poang, aven om de ar korrekta.

• Antalet bonuspoang till tentamen ar antalet poang pa denna dugga dividerat med 4.

• Endast icke-symboliska miniraknare ar tillatna.

Lycka till!

2



2013-02-049:00–13:00

1. Compute: (4 p) 4 −2 02 −1 3−3 5 1

· 0 0 2

0 −1 03 0 0

−1

·

32−2

× 4

01

.

2. Determine a ∈ R so that the following matrix is not invertible:4 −1 −2 11 0 3 1−1 2 0 2

2 5 a −1

(4 p)

3. Compute two solutions of the following equation:

X ·(

2 16 3

)=

(−6 −310 5

)In other words, find two 2× 2 matrices X such that this equation is true. (4 p)

4. Write (−3 −1

2 4

)as a product of elementary matrices. (2 p)

5. Solve the following linear system: (4 p)

x1 + 3x3 + 5x4 + 6x5 = 3−2x1 + 2x2 − 6x3 − 14x4 − 8x5 = −2

x1 + 2x2 + 3x3 + 3x4 + 10x5 = 7

6. Let u = (2,−4, 1,−1) and a = (5, 1,−1,−3).

Express u as sum of two vectors w1 and w2,where w1 is a scalar multiple of a and w2 is orthogonal to a. (4 p)

7. Let A = (1, 2,−1), B = (0, 1, 3), C = (2, 0, 1), D = (0, 0, 1), E = (2, 3,−1).

Consider the plane P containing the points A, B, C, and the line L

containing the points D, E.

(a) Write the line in vector equation form. (1 p)

(b) Write the plane in vector equation form. (1 p)

(c) Compute the intersection of L and P . (2 p)

8. (a) Show that the planes with point-normal equations4x− 2y − 3z = 6 and −6x + 3y + 9

2z = 2 are parallel. (2 p)(b) Compute the distance between the two planes. (2 p)


• A Swedish version of the pre-exam is available on the opposite side of this sheet. Solutionsmay be written in Swedish or English.



• Calculators are allowed.

Good luck!

1



2013-02-049:00–13:00

1. Berakna: (4 p) 4 −2 02 −1 3−3 5 1

· 0 0 2

0 −1 03 0 0

−1

·

32−2

× 4

01

.

2. Bestam a ∈ R sa att foljande matris inte ar inverterbar: (4 p)4 −1 −2 11 0 3 1−1 2 0 2

2 5 a −1

3. Berakna tva losningar till foljande ekvation:

X ·(

2 16 3

)=

(−6 −310 5

)Med andra ord, hitta tva 2× 2-matriser X sadana att likheten ar sann. (4 p)

4. Skriv (−3 −1

2 4

)som en produkt av elementara matriser. (4 p)

5. Los foljande linjara system: (4 p)

x1 + 3x3 + 5x4 + 6x5 = 3−2x1 + 2x2 − 6x3 − 14x4 − 8x5 = −2

x1 + 2x2 + 3x3 + 3x4 + 10x5 = 7

6. Lat u = (2,−4, 1,−1) och a = (5, 1,−1,−3).

Skriv u som en summa av tva vektorer w1 och w2, dar w1 ar en multipel av a,och w2 ar ortogonal mot a. (4 p)

7. Lat A = (1, 2,−1), B = (0, 1, 3), C = (2, 0, 1), D = (0, 0, 1), E = (2, 3,−1).

Lat P vara det plan som innehaller punkterna A, B och C,

och lat L vara det linje som innehaller punkterna D och E.

(a) Skriv linjen pa vektorekvationsform. (1 p)

(b) Skriv planet pa vektorekvationsform. (1 p)

(c) Bestam skarningen mellan linjen L och planet P . (2 p)

8. (a) Visa att de tva planen med punkt-normalekvationerna4x− 2y − 3z = 6 och −6x + 3y + 9

2z = 2 ar parallella. (2 p)(b) Berakna avstandet mellan planen. (2 p)





• Miniraknare ar tillatna.

Lycka till!

2



2013-05-029:00–13:00

1. Let

A =

(2 −1−6 3

), B =

(2 00 3

), C =

(1 −4 0−2 3 2

).

Compute each of the following terms, if it is defined. If is not defined, explain shortly why.

(a) A−1. (1 p)

(b) B−3. (1 p)

(c) C ·AT . (1 p)

(d) CT ·B. (1 p)

2. Compute the determinant of the matrix

4 −1 −2 12 −3 2 0−1 5 2 −1

4 −1 −1 1

(4 p)

3. Write

(−2 4−3 5

)as a product of elementary matrices. (4 p)

4. For which a ∈ R has the following linear system zero, one or infinitely many solutions? (4 p)

(4 p)

x1 + x2 + ax3 = 23x1 + 4x2 − 2x3 = a2x1 + 3x2 − x3 = 1

5. Compute the inverse of the matrix

−3 5 −21 −3 13 −6 2

6. Let A = (1, 0, 3), B = (2, 0, 3), C = (2, 1, 3), D = (2, 3, 2), E = (2, 4, 3), F = (4, 3, 3).

Consider the plane P1 containing the points A, B, C, and the plane P2 containingthe points D, E, F .

(a) Write the plane P1 in vector equation form. (1 p)

(b) Write the plane P2 in vector equation form. (1 p)

(c) Compute the intersection of P1 and P2. (2 p)

7. Determine a ∈ R so that u = (3, 2, a) and v = (3,−1, 2) are orthogonal. (2 p)

8. Let u = (0, 4,−2), v = (2,−1, 1). Determine a unit vector which is orthogonal to u and v. (2 p)

9. Compute the distance of the point A = (2, 3) to the line y = 2x− 3. (2 p)

10. Compute the area of the triangle spanned by the vectors u = (1, 2, 3) and v = (−2, 1,−1). (2 p)


• A Swedish version of the pre-exam is available on the opposite side of this sheet. Solutionsmay be written in Swedish or English.




Good luck!

1



2013-05-029:00–13:00

1. Lat

A =

(2 −1−6 3

), B =

(2 00 3

), C =

(1 −4 0−2 3 2

).

Berakna foljande uttryck, om de ar definierad. Om nagot uttryck inte ar definierat,forklara kort varfor.

(a) A−1. (1 p)

(b) B−3. (1 p)

(c) C ·AT . (1 p)

(d) CT ·B. (1 p)

2. Berakna determinanten till matrisen

4 −1 −2 12 −3 2 0−1 5 2 −1

4 −1 −1 1

(4 p)

3. Skriv

(−2 4−3 5

)som en produkt av elementara matriser. (4 p)

4. For vilka vardena pa a ∈ R har det linjara systemet noll, en eller oandligt manga losningar?x1 + x2 + ax3 = 2

3x1 + 4x2 − 2x3 = a2x1 + 3x2 − x3 = 1

5. Berakna inversen till matrisen

−3 5 −21 −3 13 −6 2

6. Lat A = (1, 0, 3), B = (2, 0, 3), C = (2, 1, 3), D = (2, 3, 2), E = (2, 4, 3), F = (4, 3, 3).

Lat P1 vara det plan som innehaller punkterna A, B, C, och lat P2 vara det plan sominnehaller punkterna D, E, F .

(a) Skriv planet pa P1 vektorekvationsform. (1 p)

(b) Skriv planet pa P2 vektorekvationsform. (1 p)

(c) Bestam skarningen mellan planen P1 och P2. (2 p)

7. Bestam a ∈ R sa att u = (3, 2, a) och v = (3,−1, 2) ar ortogonala. (2 p)

8. Lat u = (0, 4,−2), v = (2,−1, 1). Hitta en enhetsvektor w ∈ R3 som ar ortogonalmot bade u och v. (2 p)

9. Berakna avstandet fran punkten A = (2, 3) till linjen y = 2x− 3. (2 p)

10. Berakna arean av den triangle som bildas av vektoren u = (1, 2, 3) och v = (−2, 1,−1). (2 p)






Lycka till!

2


Exam in mathematicsLinear algebra, part I

2014-01-319:00–13:00

1. Let

A =

(5 18 2

), B =

(2 −4 2−1 2 5

), u = (2,−3, 2), v = (1,−2, 4).

Compute each of the following terms, if it is defined. If it is not defined, explain shortly why.

(a) A−2, (b) B4, (c) A ·B, (d) B ·AT , (e) u · v, (f) u× v, (g) B + u, (h) ‖u + v‖.(Each part: 0.5 p)

2. Compute

det

1 0 0

8 2 06 −3 −2

· 4 5 8

0 2 40 0 −3

· −1 0 0

0 1/2 00 0 3

−2

(4 p)

3. Compute the inverse of

−1 2 31 1 4−1 1 1

with the adjoint method. (4 p)

(Solutions using other methods: 2 p)

4. For which a ∈ R has the following linear system zero, one or infinitely many solutions? (4 p)

x1 + 2x2 + x3 = 2−2x1 + 2x2 + a2x3 = a

x1 − 2x2 − x3 = 0

5. Let A = (−5, 1, 3), B = (−2, 6,−3), C = (−4, 3, 4), D = (−2, 7, 6), E = (−1, 8,−2).

Consider the plane P containing the points A, B, C, and the line L

containing the points D, E.

(a) Write the line in vector equation form. (1 p)

(b) Write the plane in vector equation form. (1 p)

(c) Compute the intersection of L and P . (2 p)

6. Let u = (−1,−3, 4), v = (2, 0,−1) and w = (3,−2, 1). Consider the parallelepiped Pspanned by these three vectors.

(a) Compute the 8 corners of P . (2 p)

(b) Compute the volume of P . (2 p)

Information regarding this exam:

• A Swedish version of the exam is available on the opposite side of this sheet. Solutions maybe written in Swedish or English.



Good luck!

1


Tentamen i matematikLinjar algebra, del I

2014-01-319:00–13:00

1. Lat

A =

(5 18 2

), B =

(2 −4 2−1 2 5

), u = (2,−3, 2), v = (1,−2, 4).

Berakna foljande uttryck, om de ar definierade. Om nagot uttryck inte ar definierat,forklara kort varfor.

(a) A−2, (b) B4, (c) A ·B, (d) B ·AT , (e) u · v, (f) u× v, (g) B + u, (h) ‖u + v‖.(Varje del: 0,5 p)

2. Berakna

det

1 0 0

8 2 06 −3 −2

· 4 5 8

0 2 40 0 −3

· −1 0 0

0 1/2 00 0 3

−2

(4 p)

3. Berakna inversen till

−1 2 31 1 4−1 1 1

med adjoint metoden. (4 p)

(Losningar med andra metoder: 2 p)

4. For vilka vardena pa a ∈ R har det linjara systemet noll, en eller oandligt mangalosningar? (4 p)

x1 + 2x2 + x3 = 2−2x1 + 2x2 + a2x3 = a

x1 − 2x2 − x3 = 0

5. Lat A = (−5, 1, 3), B = (−2, 6,−3), C = (−4, 3, 4), D = (−2, 7, 6), E = (−1, 8,−2).

Lat P vara det plan som innehaller punkterna A, B och C, lat L vara det linje som innehallerpunkterna D och E.

(a) Skriv linjen pa vektorekvationsform. (1 p)

(b) Skriv planet pa vektorekvationsform. (1 p)

(c) Bestam skarningen mellan linjen L och planet P . (2 p)

6. Lat u = (−1,−3, 4), v = (2, 0,−1) och w = (3,−2, 1). Betrakta den parallellepiped Pom bestams av dessa tre vektorer.

(a) Berakna de 8 hornen av P . (2 p)

(b) Berakna volymen av P . (2 p)

Information om tentamen:

• En engelsk version av tentamen ar tillganglig pa andra sidan av detta blad. Losningar kanskrivas pa svenska eller engelska.



Lycka till!

2


Second exam in mathematicsLinear algebra, part I

2014-02-219:00–15:00

1. Let

A =

(2 35 7

), B =

1 6−2 5

0 3

, u = (1, 2, 5), v = (−1, 3, 4).

Compute each of the following terms, if it is defined. If it is not defined, explain shortly why.

(a) A−1, (b) B2, (c) tr(A), (d) B ·AT , (e) u · v, (f) u× v, (g) B · u, (h) ‖u + v‖.(Each part: 0.5 p)

2. Compute (4 p)

det

5 4 0 0 0−3 2 0 0 0

2 0 −3 0 0−1 1 2 4 0

3 4 2 −3 −1

3. Compute the inverse of

4 4 −3−1 −1 1

1 2 3

(4 p)

4. Solve the following linear system using Cramer’s rule: (4 p)

2x1 + 4x2 + 7x3 = 2− 2x2 + 4x3 = 3

3x1 + 7x2 + 8x3 = −1

(Solutions using other methods: 2 p)

5. Let A = (−2, 3, 2), B = (−3, 0, 4), C = (3, 1, 2) and D = (1, 2− 3).

Consider the plane P containing the points A, B, C.

(a) Write the plane in vector equation form. (1 p)

(b) Write the plane in point-normal equation form. (2 p)

(c) Compute the distance between the point D and the plane P . (1 p)

6. Let A = (0, 0, 0), B = (0, 1, 1) and C = (1, a, 1). Consider the triangle T in R3

with the corners A, B and C. Determine a ∈ R so that the area of T is 1. (4 p)





Good luck!

1


Andra tentamen i matematikLinjar algebra, del I

2014-02-219:00–15:00

1. Lat

A =

(2 35 7

), B =

1 6−2 5

0 3

, u = (1, 2, 5), v = (−1, 3, 4).

Berakna foljande uttryck, om de ar definierade. Om nagot uttryck inte ar definierat,forklara kort varfor.

(a) A−1, (b) B2, (c) tr(A), (d) B ·AT , (e) u · v, (f) u× v, (g) B · u, (h) ‖u + v‖.(Varje del: 0,5 p)

2. Berakna (4 p)

det

5 4 0 0 0−3 2 0 0 0

2 0 −3 0 0−1 1 2 4 0

3 4 2 −3 −1

3. Berakna inversen till

4 4 −3−1 −1 1

1 2 3

(4 p)

4. Los det linjara systemet med Cramers regel: (4 p)

2x1 + 4x2 + 7x3 = 2− 2x2 + 4x3 = 3

3x1 + 7x2 + 8x3 = −1

(Losningar med andra metoder: 2 p)

5. Lat A = (−2, 3, 2), B = (−3, 0, 4), C = (3, 1, 2) och D = (1, 2− 3).

Lat P vara det plan som innehaller punkterna A, B, C.

(a) Skriv planet pa vektorekvationsform. (1 p)

(b) Skriv planet pa punktnormalform. (2 p)

(c) Berakna avstandet fran punkten D till planet P . (1 p)

6. Lat A = (0, 0, 0), B = (0, 1, 1) och C = (1, a, 1). Betrakta trianglen T i R3

definierad av hornen A, B och C. Berakna a ∈ R sa att ytan av trianglen T ar 1. (4 p)





Lycka till!

2


Third exam in mathematicsLinear algebra, part I

2014-03-229:00–15:00

1. Let u, v ∈ R3 and A,B ∈ R2,2. State whether the following assertions are true in general

or not. If yes, give an example. If not, give a counterexample. (All examples or

counterexamples should contain no zero elements.)

(a) u× v = v × u. (1 p)

(b) ‖u + v‖ = ‖u‖+ ‖v‖. (1 p)

(c) Let A be invertible.(A2)−1

=(A−1

)2. (1 p)

(d) (A ·B)T

= AT ·BT . (1 p)

2. For which a ∈ R is the matrix

2 −2 −13 −4 2−2 1 a

invertible? Explain your answer. (4 p)

3. Let A =

−1 −2 6−1 −1 2

4 4 −7

and b =

12−1

a) Compute the inverse of A. (3 p)

b) Use the result of a) to solve the linear system Ax = b. (1 p)(Solutions using other methods: 0.5 p)

4. Let A = (1, 2,−1), B = (2, 3, 1) and C = (−1, 3, 4). Consider the triangle T in R3

with the corners A, B and C. Compute the area of the triangle T . (4 p)

5. Let 2x + 4y − 4z = 4 be the point-normal equation form for the plane P1

and −3x− 6y + 6z = 2 be the point-normal equation for the plane P2.

(a) Write the plane P1 in vector equation form. (2 p)

(b) Compute the intersection of P1 and P2. (2 p)

6. a) Formulate the theorem about the parallelogram equation for vectors. (1 p)

b) Prove the theorem about the parallelogram equation for vectors. (3 p)





Good luck!

1


Tredje tentamen i matematikLinjar algebra, del I

2014-03-229:00–15:00

1. Lat u, v ∈ R3 och A,B ∈ R2,2. Avgor om foljande uttalanden ar sanna i allmanhet

eller inte. Om ja, ge ett exempel. Om nej, ge ett motexempel. (Alla exampel

eller motexempel ska inte innehalla nagra nollelement.)

(a) u× v = v × u. (1 p)

(b) ‖u + v‖ = ‖u‖+ ‖v‖. (1 p)

(c) Lat A vara inverterbar.(A2)−1

=(A−1

)2. (1 p)

(d) (A ·B)T

= AT ·BT . (1 p)

2. For vilka varden pa a ∈ R ar matrisen

2 −2 −13 −4 2−2 1 a

inverterbar? Forklara ditt svar. (4 p)

3. Lat A =

−1 −2 6−1 −1 2

4 4 −7

och b =

12−1

a) Berakna inversen till A. (3 p)

b) Anvand resultatet av a) for att losa linjara systemet Ax = b. (1 p)(Losningar med andra metoder: 0.5 p)

4. Lat A = (1, 2,−1), B = (2, 3, 1) och C = (−1, 3, 4). Betrakta triangeln T i R3

definierad av hornen A, B och C. Berakan arean av triangeln T . (4 p)

5. Lat 2x + 4y − 4z = 4 vara punktnormalform av planet P1

och −3x− 6y + 6z = 2 vara punktnormalform av planet P2.

(a) Skriv planet P1 pa vektorekvationsform. (2 p)

(b) Bestam skarningen mellan P1 och P2. (2 p)

6. a) Formulera satsen om parallelogramekvationen for vektorer. (1 p)

b) Bevisa satsen om parallelogramekvationen for vektorer. (3 p)





Lycka till!

2

Documents

UME A UNIVERSITY 2012-02-07 9:00{13:00 · (b) Ber akna avst andet mellan planen. (2 p) Information r orande denna dugga: En engelsk version av duggan ar tillg anglig p a andra sidan