Prepared By ASCOL CSIT 2070 Batch

IOST, TU

Tribhuvan University

Institute of Science and Technology

2065

囎

Bachelor Level/First Year/ Second Semester/ Science Full Marks: 80

Computer Science and Information Technology Pass Marks: 32

(MTH.155 – Linear Algebra) Time: 3hours

Candidates are required to give their answers in their own words as far as practicable.

The figures in the margin indicate full marks.

Attempt all questions:

Group A (10 x 2 = 20)

1. Illustrate by an example that a system of linear equations has either equations has either exactly one

solution or infinitely many solutions.

2. When is a linear transformation invertible?

3. Solve the system ぬ捲怠 髪 ね捲態 噺 ぬ┸ の捲怠 髪 は捲態 噺 ばby using the inverse of the matrix 畦 噺 峙ぬ ねの は峩┻

4. State the numerical importance of determinant calculation by row operation.

5. State Cramer’s rule for an invertible n x n matrix A and vector 決 香 迎津 to solve the system 畦捲 噺 決. Is

this method efficient from computational point of view?

6. Determine if 岶懸怠┸ 懸態懸戴岼 is basis for R3, where 懸怠 噺 煩などど晩┸ 懸態 噺 煩ななど晩┸ 懸戴 噺 煩ななな晩┻

7. Determine if 激 噺 煩 な伐ね晩 is a Nul(A) for 畦 噺 煩 伐の 伐ぬは 伐に ど伐ぱ ね な 晩┻8. Show that 7 is an eigen value of 畦 噺 峙な はの に峩┻9. If 鯨 噺 版憲怠┸ 橋 橋 橋 橋 ┸ 憲椎繁 is an orthogonal set of nonzero vectors in R

2, show S is linearly

independent and hence is a basis for the subspace spanned by S.

33

1CSc. MTH. 155-2065 囎

IOST, TU

10. Let 激 噺 嫌喧欠券岶捲怠┸ 捲態岼 where 捲怠 噺 煩伐のな 晩 and 捲態 噺 煩 ね伐な晩┻ Their construct orthogonal basis for W.

Group B (5 x 4 = 20)

11. Determine if the given set is linearly dependent:

a) 煩なばは晩 ┸ 煩どひ晩 ┸ 煩ぬな晩 ┸ 煩ねなぱ晩 b) 崛伐にねはなど崑 ┸ 崛 ぬ伐は伐ひなの崑12. Find the 3 x 3 matrix that corresponds to the composite transformation of a scaling by 0.3, a rotation

of 900, and finally a translation that adds (-0.5, 2) to each point of a figure.

OR

Describe the Leontief Input-Output model for certain economy and derive formula for (I-C)-1

, where

symbols have their usual meanings.

13. Find the coordinate vector 岷隙峅喋 of a x relative to the given basis 稽 噺 岶決怠┸ 決態岼, where決怠 噺 峙 な伐ぬ峩┸ 決態 噺 峙 に伐の峩 ┸ 捲 噺 峙伐にな 峩┻14. Let 畦 噺 峙ね 伐ぱね ぱ峩┸ 決怠 噺 峙ぬに峩┸ 決態 噺 峙にな峩 and basis 稽 噺 岶決怠┸ 決態岼. Find the B-matrix for the

transformation 捲 蝦 畦捲 with 鶏 噺 岶決怠┸ 決態岼┻15. Let u and v be non-zero vectors in R

3 and the angle between them be. Then prove that憲┻ 懸 噺 押憲押押懸押 叶┸

where the symbols have their usual meanings.

Group C (5 x 8 = 40)

16. Let 劇┺ 迎津 蝦 迎陳 be a linear transformation. Then T is one-to-one if and only if the equation T(x) = 0

has only the trivial solution, prove the statement.

OR

Let 畦 噺 煩 な 伐ぬぬ伐な ば 晩 ┸ 憲 噺 峙 に伐な峩 ┸ 決 噺 煩 ぬ伐の晩 ┸ 潔 噺 煩ぬに晩 and define 劇┺ 迎態 蝦 迎戴 by 劇岫捲岻 噺畦捲. Then

a) Find T(u)

b) Find an 捲 樺 迎態 whose image under T is b.

c) Is there more than one x whose image under T is b?

d) Determine if c is the range of T.

2

2

2

2

5

55

1CSc. MTH. 155-2065 囎

IOST, TU

17. Compute the multiplication of partitioned matrices for

畦 噺 頒 伐ぬ な ど 伐ねな の 伐に 伐など ね 伐に ば 伐な番 欠券穴 稽 噺 琴欽欽欽欽欣 は ね伐に な伐ぬ ば伐なの 筋禽禽

禽禽禁┻18. What do you mean by change of basis in R

n? Let 決怠 噺 峙 な伐 峩┸ 決態 噺 峙伐にね 峩┸ 潔怠 噺 峙伐ばひ 峩┸ 潔態 噺 峙伐のば 峩,

and consider the bases for R2 given by 稽 噺 岶決怠┸ 決態岼 and 系 噺 岶潔怠┸ 潔態岼.

a) Find the change of coordinate matrix from C to B.

b) Find the change of coordinate matrix from B to C.

OR

Define vector spaces, subspaces, basis of vector space with suitable examples. What do you mean by

linearly independent set and linearly dependent set of vectors?

19. Diagonalize the matrix 畦 噺 煩伐な ね 伐に伐ぬ ね ど伐ぬ な 晩┸ if possible.

20. Find the equation 検 噺 紅待 髪 紅怠捲 of the least squares line that best fits the data points (2, 1), (5, 2),

(7, 3), (8, 3). What do you mean by least squares lines?

2

2

33

3

3

Prepared By Ascol CSIT 2070 Batch

IOST, TU

Tribhuvan University

Institute of Science and Technology

2066

囎

Bachelor Level/First Year/ Second Semester/ Science Full Marks: 80

Computer Science and Information Technology Pass Marks: 32

(MTH.155 – Linear Algebra) Time: 3hours

Candidates are required to give their answers in their own words as far as practicable.

The figures in the margin indicate full marks.

Attempt all questions:

Group A (10 x 2 = 20)

1. When is system of linear equation consistent or inconsistent?

2. Write numerical importance of partitioning matrices.

3. How do you distinguish singular and non-singular matrices?

4. If A and B are n x n matrices, then verify with an example that det(AB) = det(A)det(B).

5. Calculate the area of the parallelogram determined by the columns of畦 噺 峙に はの な峩┻6. Determine if 拳 噺 煩 なぬ伐ね晩 is Nul(A), where, 畦 噺 煩 ぬ 伐の 伐ぬは 伐に ど伐ぱ ね な 晩┻7. Determine if {v1, v2, v3} is a basis for 3

, where 懸怠 噺 煩などど晩┸ 懸態 噺 煩ななど晩┸ 懸戴 噺 煩ななな晩┻8. Find the characteristic polynomial for the eigen values of the matrix 峙 に な伐な ね峩┻9. Let 懸王 噺 岫な┸ 伐に┸ に┸ ど岻. Find a unit vector 憲屎王 in the same direction as 懸王.

10. Let 版憲怠┸ 橋 橋 橋 ┸ 憲椎繁 be an orthogonal basis for a subspace W of Rn. Then prove that for each 検香激,

the weights in 検 噺 潔怠憲怠 髪 橋 橋 橋 髪 潔椎憲椎 are given by潔珍 噺 検┻ 憲珍憲珍┻ 憲珍 岫倹 噺 な┸ 橋 橋 橋 ┸ 喧岻

1CSc. MTH. 155-2066 囎

IOST, TU

Group B (5 x 4 = 20)

11. Prove that any set 版懸怠┸ 橋 橋 橋 ┸ 懸椎繁 in Rn is linearly dependent if 喧 伴 券.

12. Consider the Leontief input – output model equation 捲 噺 潔捲 髪 穴, where the consumption matrix is系 噺 煩┻ のど ┻ ねど ┻ にど┻ にど ┻ ぬど ┻ など┻ など ┻ など ┻ ぬど晩┻Suppose the final demand is 50 units of manufacturing, 30 units of agriculture, 20 units for services.

Find the production level x that will satisfy the demand.

13. What do you mean by basis of a vector space? Find the basis for the row space of

畦 噺 頒伐に 伐の ぱ ど 伐なばな ぬ 伐の な 5 ぬな ななば 伐なひ伐なぬ ば の な伐ぬ 番

OR

State and prove the unique representation theorem for coordinate systems.

14. What do you mean by eigen values, eigen vectors and characteristic polynomial of a matrix? Explain

with suitable examples.

15. Define the Gram-Schmidt process. Let W=span{x1, x2}, where 捲怠 噺 煩ぬはど晩┸ 捲態 噺 煩なにに晩┻ Then

construct an orthogonal basis {v1, v2} for w.

Group C (5 x 8 = 40)

16. Given the matrix 煩ど ぬ 伐はぬ 伐ば ぱぬ 伐ひ なに は 伐の伐の ひ伐ひ なの晩┸discuss the for word phase and backward phase of the row reduction algorithm.

17. Find the inverse of 煩 な ど 伐に伐ぬ な ねに 伐ぬ ね 晩┸ if it exists, by using elementary row reduce the augmented

matrix.

1CSc. MTH. 155-2066 囎

IOST, TU

18. What do you mean by change of basis in Rn ? Let 決怠 噺 峙伐ひな 峩 ┸ 決態 噺 峙伐の伐 峩 ┸ 潔怠 噺 峙 な伐ね峩 ┸ 潔態 噺 峙 ぬ伐の峩┸

and consider the bases for R2 given by B={b1, b2} and C={c1, c2}. Find the change of coordinates

matrix from B to C.

19. Diagonalize the matrix 煩 に に 伐なぬ 伐な伐な 伐に に 晩, if possible

OR

Find the eigen value of 畦 噺 峙ど┻のど 伐ど┻はどど┻ばの な┻な 峩, and find a basis for each eigen space.

20. Find a least-square solution for Ax = b with 畦 噺 崛 伐は伐にば 崑 ┸ 決 噺 崛伐なには 崑┻ What do you mean by least

squares problems?

OR

Define a least-squares solution of Ax = b, prove that the set of least squares solutions of Ax = b

coincides with the non-empty set of solutions of the normal equations ATAx = A

Tb.

1 1

1111

1

1

Tribhuvan University

Institute of Science and Technology

2067

Bachelor Level/ First Year/ Second Semester/ Science Full Marks: 80

Computer Science and Information Technology (MTH 155) Pass Marks: 32

(Linear Algebra) Time: 3 hours.

Candidates are required to give their answers in their own words as for as practicable.

The figures in the margin indicate full marks.

Attempt all questions:

Group A (10x2=20)

1. Illustrate by an example that a system of linear equations has either no solution or exactly

one solution.

2. Define singular and nonsingular matrices.

3. Using the Invertible matrix Theorem or otherwise, show that

[

]

is invertible.

4. What is numerical drawback of the direct calculation of the determinants?

5. Verify with an example that et ( ) et ( ) et ( ) for any n x n matrices A and B.

6. Find a matrix A such that ( ).

{[

] : }.

7. Define subspace of a vector with an example.

8. Are the vectors;

0 1 an

0 1 eigen vectors of 0

1?

9. Find the distance between vectors u ( ) and v ( ). Define the distance between

two vectors in .

10. Let w = span {x1, x2}, where [ ] [

].

Then construct orthogonal basis for w.

Group B (5x4=20)

11. If a set s = {v1, v2 vp} in n contains the zero vector, then prove that the set is linearly

dependent. Determine if the set

[ ] [ ] [

]

is linearly dependent.

12. Given the Leontief input-output model x = Cx + d, where the symbols have their usual

meanings, consider any economy whose consumption matrix is given by

[

]

Prepared By ASCOL CSIT 2070 Batch

0

0

0

0

IOST, TU

A =

=

=

= =

= =

Suppose the final demand is 50 units for manufacturing 30 units for agriculture, 20 units for

services. Find the production level x that will satisfy this demand.

13. Define rank of a matrix and state Rank Theorem. If A is a 7 x 9 matrix with a

two-dimensional null space, find the rank of A.

14. Determine the eigen values and eigen vectors of 0

1 in complex numbers.

OR

Let 0

1 0 1 0

1 and basis B = { }.

Find the B-matrix for the transformation x x with , 1, b2].

15. Let u and v be nonzero vectors in an the angle etween them e θ then prove that

u v ‖u‖ ‖v‖ os θ,

where the symbols have their usual meanings.

16. Determine if the following homogeneous system has a nontrivial solution. Then describe the

solution set.

.

17. An n x n matrix A is invertible if and only if A is row equivalent to n , and in this case, any

sequence of elementary row operations that reduces A to n also transform n x m into .

Use this statement to find the inverse of the matrix [

], if exists.

18. What do you mean by basis change? Consider two bases B = {b1, b2} and c = {c1, c2} for a

vector space V, such that and . Suppose , - 0 1 i.e., x = 3b1 +

b2. Find , -C.

OR

Define basis of a subspace of a vector space.

Let v [ ] v [

] v [

], where v v v , and let span *v v v +.

Show that span *v v v + span *v v + and find a basis for the subspace H.

19. Diagonalize the matrix [

], if possible.

20. What do you mean by least-squares lines? Find the equation of the least-

squares line that fits the data points (2, 1), (5, 2), (7, 3), (8, 3).

OR

Find the least-squares solution of Ax = b for

[

] [

].

00

0 0

0

0

0

IOST, TU

A =

A

A

During our semester csitnepal.com don’t allow us to download so we have collect it from seniors as a hard copy and

scan it as jpg format so we had upload it……………..

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Prepared By ASCOL CSIT 2070 (2068 part 2)

20. Find the equation 𝑦 = 𝛽0 + 𝛽1𝑥 for the least squares line that best fits the data points (2, 0), (3, 4), (4, 10), (5, 16).

ASCOL CSIT 2070 Batch

ASCOL CSIT 2070 Batch

ASCOL CSIT 2070