Mathematics II - Unit Wise Questions

Unit 1: Linear Equations in Linear Algebra
186 Questions

1. Define system of linear equations. When a system of equations is consistent? Determine if the system 

    -2x1-3x2+4x3 = 5

    x2-2x3 = 4

    x1+3x2-x3 = 2

is consistent.            [1+1+8]

10 marks | Asked in 2078

1. What is pivot position? Apply elementary row operation to transform the following matrix first into echelon form and then into reduced echelon form:


10 marks | Asked in Model Question

1. Illustrate by an example that a system of linear equations has either equations has either exactly one solution or infinitely many solutions.

2 marks | Asked in 2065

1. What is a system of linear equations? When the system is consistent and inconsistent?

2 marks | Asked in 2071

1. Write down the conditions for consistent of non- homogenous system of linear equations.

2 marks | Asked in 2068

1. When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve: x + y + z = 4, x + 2y + 2z = 2, 2x + 2y + z = 5.

10 marks | Asked in 2075(New Course)

1. When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve the system of equations: x - 2y = 5, -x + y + 5z = 2, y + z = 0

10 marks | Asked in 2076

1. Why the system x1 - 3x2 = 4; -3x1 + 9x2 = 8 is consistent? Give the graphical representation?

2 marks | Asked in 2070

1. What are the criteria for a rectangular matrix to be in echelon form?

2 marks | Asked in 2074

1. What do you mean by linearly independent set and linearly dependent set of vectors?

2 marks | Asked in 2069

1. Define linear combination of vectors. When the vectors are linearly dependent and independent?

2 marks | Asked in 2072

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

2 marks | Asked in 2066

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

one solution.

2 marks | Asked in 2067

2. Define singular and nonsingular matrices.

2 marks | Asked in 2067

2. Define linear combination of vectors. If v1, v2,v3 are vectors, Write the linear combination of 3v1 - 5v2 + 7v3 as a matrix times a vector.

2 marks | Asked in 2070

2. Write numerical importance of partitioning matrices.

2 marks | Asked in 2066

2. prove that (a) (AT)T = A  (b) (A + B)T = AT + BT , Where A and B denote matrices whose size are appropriate for the above mentioned operations.

2 marks | Asked in 2074

2. Define linear transformation between two vector spaces.

2 marks | Asked in 2072

2. What is meant by independent of vectors?

2 marks | Asked in 2068

2. When is a linear transformation invertible?

2 marks | Asked in 2065

2. Define linearly dependent and independent vectors. If (1, 2) and (3, 6) are vectors then the vectors are linearly dependent or independent?

2 marks | Asked in 2071

2. Verify that  is an eigen vector of 
2 marks | Asked in 2069

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

2 marks | Asked in 2067

3. Show that the matrix is not invertible.

2 marks | Asked in 2072

3. Define square matrix. Can a square matrix with two identical columns be invertible? Why or why not?
2 marks | Asked in 2074

3. Solve the system 

3x1 + 4x2 = 3,               5x1 + 6x2 = 7

by using the inverse of the matrix 

2 marks | Asked in 2065

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

2 marks | Asked in 2066

3. Is  invertible matrix?

2 marks | Asked in 2070

3. What is normal form of a matrix?

2 marks | Asked in 2068

3. Define invertible matrix transformation. 

2 marks | Asked in 2071

3. What do you mean by consistent equations? Give suitable examples.

2 marks | Asked in 2069

4. Let A and B be two square matrices. By taking suitable examples, show that even though AB and BA may not be equal, it is always true that detAB = detBA.

2 marks | Asked in 2074

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

2 marks | Asked in 2065

4. Let S be the parallelogram determined by the vectors b1 = (1, 3) and b2 = (5, 1) and let  Compute the area of the image S under the mapping 

2 marks | Asked in 2071

4. Define invertible linear transformation.

2 marks | Asked in 2070

4. What do you mean by change of basis in Rn?

2 marks | Asked in 2069

4. Define nonsingular linear transformation with suitable example.

2 marks | Asked in 2068

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

2 marks | Asked in 2067

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

2 marks | Asked in 2066

4. Define invertible matrix transformation.

2 marks | Asked in 2072

5. Determine the column of the matrix A are linearly independent, where

    

5 marks | Asked in 2078

5. State Cramer’s rule for an invertible n x n matrix A and vector  to solve the system Ax = b. Is this method efficient from computational point of view?

2 marks | Asked in 2065

5. Calculate the area of the parallelogram determined by the columns of 

2 marks | Asked in 2066

5. Find the dimension of the vector spanned by (1,1,0) and (0,1,0).

2 marks | Asked in 2069

5. Let S be the parallelogram determined by the vectors b1 = (1, 3) and b2 = (5, 1) and let Compute the area of the image S under the mapping 

2 marks | Asked in 2070

5. Show that the matrices do not commute.

2 marks | Asked in 2071

5. Consider the matrix  as a linear mapping. Write the corresponding co-ordinate equations.

2 marks | Asked in 2068

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

2 marks | Asked in 2067

5. Let S be the parallelogram determined by the vectors b1 = (1,3) and b2 = (5,1) and let . Compute the area of the image S under the mapping 

2 marks | Asked in 2072

5. Using Cramer's rule solve the following simultaneous equations:

5x + 7y = 3

2x + 4y = 1

2 marks | Asked in 2074

6. Define vector space.

2 marks | Asked in 2071

6. Find a matrix A such that col(A).


2 marks | Asked in 2067

6. Define subspace of a vector space.

2 marks | Asked in 2072

6. When is a linear transformation invertible.

2 marks | Asked in 2069

6. Determine if {v1, v2, v3} is basis for R3, where    

2 marks | Asked in 2065

7. Determine if {v1, v2, v3} is a basis for  , where 

2 marks | Asked in 2066

6. Define vector space with suitable examples.

2 marks | Asked in 2074

6. Define vector space.

2 marks | Asked in 2070

5. Change into reduce echelon form of the matrix 

5 marks | Asked in 2075(New Course)

7. Let W be the set of all vectors of the form , where b and c are arbitrary. Find vector u and v such that W = Span {u, v}.

2 marks | Asked in 2074

7. Find the rank of AB where 

2 marks | Asked in 2069

7. Show that the entries in the vector x = (1, 6) are the coordinates of x relative to the standard basis (e1, e2).

2 marks | Asked in 2070

7. Determine if w = (1, 3, -4) is a Nul A, where 

2 marks | Asked in 2071

7. Define subspace of a vector with an example.

2 marks | Asked in 2067

7. Determine if  is a Nul(A) for 

2 marks | Asked in 2065

6. Determine if  is Nul(A), where, 

2 marks | Asked in 2066

7. If and let u = (5, 3, 2), then show that u is in the Nul A.

2 marks | Asked in 2072

8. Show that 7 is an eigen value of 

2 marks | Asked in 2065

8. Is an Eigen value of  

2 marks | Asked in 2070

8. What are necessary and sufficient conditions for a matrix to be invertible?

2 marks | Asked in 2074

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

2 marks | Asked in 2068

8. Are the vectors;  eigen vectors of  

2 marks | Asked in 2067

8. If u = (6, -5) is an eigen vector of ?

2 marks | Asked in 2072

8. Is u = (3, -2) is an eigen value of  ?

2 marks | Asked in 2071

8. Find the characteristic polynomial for the eigen values of the matrix 

2 marks | Asked in 2066

8. Is  an Eigen value of  ?

2 marks | Asked in 2069

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

two vectors in  Rn

2 marks | Asked in 2067

9. Let  Find a unit vector  in the same direction as 

2 marks | Asked in 2066

7. Show that {(1, 1), (-1,0)} form a bias for R2.

2 marks | Asked in 2068

9. Determine whether the pair of vectors are orthogonal or not?

2 marks | Asked in 2074

9. If S = {u1,... .... ... ... , up} is an orthogonal set of nonzero vectors in R2, show S is linearly independent and hence is a basis for the subspace spanned by S.

2 marks | Asked in 2065

9. Find the inner product of (2, -5, -1) and (3, 2, -3).

2 marks | Asked in 2071

9. Define kernel and image of linear transformation.

2 marks | Asked in 2069

9. Find the inner product of (1, 2, 3) and (2, 3, 4).

2 marks | Asked in 2070

9. Find the unit vector u of v = (1, -2, 2, 0) along the direction of v.

2 marks | Asked in 2072

10. Let w = span {x1, x2}, where  Then construct orthogonal basis for w.

2 marks | Asked in 2067

10. Find the norm between the vectors u = (1, 2, 3, 4) and v = (0, 1, 2, 3).

2 marks | Asked in 2071

10. Let {u1,... ... ... up}be an orthogonal basis for a subspace W of Rn. Then prove that for each ,the weights in y = c1u1 + ... ... ... + cpup are given by


2 marks | Asked in 2066

8. Let be a linear transformation defined by T(x, y) = (x + y, y). Find Ker T. 

2 marks | Asked in 2068

10. Compute the norm between the vectors 4 = (7, 1) and v = (3, 2).

2 marks | Asked in 2070

10. Let W = span{x1, x2} where  and Their construct orthogonal basis for W.

2 marks | Asked in 2065

10. What do you understand by least square line? Illustrate.

2 marks | Asked in 2074

10. Find the norm vector  v = (1, -2, 3, 0).

2 marks | Asked in 2072

10. What is meant by Discrete dynamical system? Give suitable example.

2 marks | Asked in 2069

9. If  λ is an eigen values of matrix A, find the eigen values of A-1.

2 marks | Asked in 2068

11. Let  be the linear transformation defined by T(x, y, z) = (x, y, x - 2y). Find a basis and dimension of (a) Ker T (b) Im T

4 marks | Asked in 2069

11. Determine if the given set is linearly dependent:



4 marks | Asked in 2065

11. Let  and  . Find the images under T of 

4 marks | Asked in 2072

11. A linear transformation  is defined by . Find the image of T of  

4 marks | Asked in 2070

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

4 marks | Asked in 2067

11. Let , u = (1, 0, -3) and v = (5, -1, 4), If   defined by T(x) = Ax, find T (u) and T (v).

4 marks | Asked in 2071

11. Prove that any set{v1,... ... ... ... , v2} in Rn is linearly dependent if p > n.

4 marks | Asked in 2066

11. What are the criteria for a transformation T to be linear? If is defined by T(x) = 3x,

Show that T is a linear transformation. Also give a geometric description of the transformation



4 marks | Asked in 2074

10. Let u = (1,2,-1,3) and v = (3,0,2,-2). Compute the inner product (u, u + v).

2 marks | Asked in 2068

12. Show that the following vectors are linearly independent:(1, 1, 2),(3, 1, 2),(0, 1, 4).

4 marks | Asked in 2069

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.

4 marks | Asked in 2065

12. Consider the Leontief input – output model equation x = cx + d, where the consumption matrix is

 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.


4 marks | Asked in 2066

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 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.


4 marks | Asked in 2067

12. If  compute (Ax)T, xTAT and xxT. Can you compute xTAT

4 marks | Asked in 2070

12. Let  , show that det (A + B) = det A + det B if a + d = 0.
4 marks | Asked in 2071

12. Find the determinant of 

4 marks | Asked in 2072

12. Prove that if A is an invertible matrix, then so is AT, and the inverse of AT is the transpose of A-1.

4 marks | Asked in 2074

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

OR

State and prove the unique representation theorem for coordinate systems.



4 marks | Asked in 2066

13. Find the coordinate vector [X]B of a x relative to the given basis B = {b1, b2}, where



4 marks | Asked in 2065

13. Define subspace of a vector space V. Given v1 and v2 in a vector space V, let H = span {v1, v2}. Show that H is a subspace of V.

OR

If  is a basis for a vector space V and x is in V, define the coordinate of x relative to the basis  . Let Then is a basis for H = Span {v1, v2}. Determine is X is in H, and if it is, find the coordinate vector of x relative to 

4 marks | Asked in 2074

13. Show that the vectors (1, 0, 0),(1, 1, 0) and (1, 1, 1) are linearly independent.

4 marks | Asked in 2072

13. If v1 and v2 are the vectors of a vector space V and H = span {v1, v2}, then show that H is a subspace of V.

4 marks | Asked in 2071

13. If b1 = (2, 1), and B = {b1, b2}, find the co-ordinate vector [x]B of x relative to B.

4 marks | Asked in 2070

13. Find the matrix representation of linear transformation defined by T(x, y) = (x + 2y) relative to the standard basis.

4 marks | Asked in 2069

11. Determine whether the following vectors in R3 are linearly dependent:

a. (1,0,1), (1,1,0),(-1,0,-1),

b. (2,1,1),(3,-2,2),(-1,2,-1).

4 marks | Asked in 2068

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.

4 marks | Asked in 2067

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

4 marks | Asked in 2066

14. Find the eigen values of  

4 marks | Asked in 2071

14. Find the eigen values of .

4 marks | Asked in 2070

14. Determine the eigen values and eigen vectors of   in complex numbers.

OR

Let and basis B = {b1, b2}.Find the B-matrix for the transformation  with P= [b1, b2].

4 marks | Asked in 2067

14. Is the set of vectors {(91, 0, 1),(0, 1, 0),(-1, 0, 1)} orthogonal? Obtain the corresponding orthonomal set R3.

4 marks | Asked in 2069

12. Investigate and interpret geometrically the transformation of the unit square whose vertices are O(0,0,1),A(1,0,1),B(0,1,1),and C(1,1,1) effected by the 3 x 3 matrix:

OR

Is the set of vectors {(),(),()} orthogonal? Obtain the corresponding orthogonal? Obtain the corresponding orthonormal set in R3.

4 marks | Asked in 2068

14. Let   and basis B = {b1, b2}.Find the B-matrix for the transformation   with P = {b1, b2}.

4 marks | Asked in 2065

14. The mapping  defined by T(a0 + a1t + a2t2) = a1 + 2a2t is a linear transformation.

a) Find the matrix for T, when is the basis {1, t, t2}.

b) Verify that [T(p)]B = [T]B[p]B for each p in P2.

4 marks | Asked in 2074

14. Find the eigen values of 

4 marks | Asked in 2072

15. Let u and v be non-zero vectors in R3 and the angle between them be  Then prove that where the symbols have their usual meanings.

4 marks | Asked in 2065

15. Let  Find a least square solution of Ax = b, and compute the associated least square error.

4 marks | Asked in 2074

15. If v1 = (3, 6, 0), v2 = (0, 0, 2) are the orthogonal basis then find the orthonal basis of v1 and v2.

OR

Find an orthogonal projection of y onto u, where y = (7, 6), u = (4, 2)

4 marks | Asked in 2072

15. Let u and v be nonzero vectors in R2 and the angle between them be θ then prove that u.v = ‖u‖ ‖v‖ cos θ,

where the symbols have their usual meanings.

4 marks | Asked in 2067

15. Show that {v1, v2, v3} is an orthogonal set, where v1 = (3,1,1), v2 = (-1, 2, 1), 

4 marks | Asked in 2070

15. Define the Gram-Schmidt process.  Let W=span{x1, x2}, where  Then construct an orthogonal basis {v1, v2} for w.

4 marks | Asked in 2066

13. In the vestor space R2, express the given vector the given vector (1,2 ) as a linear combination of the vectors (1, -1) and (0,1)

4 marks | Asked in 2068

15. Let the four vertices O(0, 0), A(1, 0), B(0, 1) and C(1, 1) of a unit square be represented by 2 x 4 matrix . Investigate and interpret geometrically the effect of pre-multiplication of this matric by the 2 x 2 matrix:

OR

State and prove orthogonality property for any two non-zero vectors in Rn.

4 marks | Asked in 2069

15. Show that (v1, v2, v3) is an orthogonal basis of R3, where 

OR

Find an orthogonal projection of y onto u, where y = (7, 6), u = (4, 2).


4 marks | Asked in 2071

14. Find the matrix representation of the linear transformation defined by T(x,y) = (x,x + 2y) relative to the basis (1,0) and (1,1).

4 marks | Asked in 2068

16. Let  be a linear transformation and let A be the standard matrix for T. Then prove that: T map Rn on to Rm if and only if the columns of A span Rm; and T is one-to-one if and only if the columns of A are linearly independent. Let T(x1, x2) = (3x1 + x2, 5x1 + 7x2, x1 + 3x2). Show that T is a one-to-one linear transformation. Does T map R2 onto R3?

8 marks | Asked in 2074

16. Determine if the following system is inconsistent.

x2 - 4x3 = 8

2x1 - 3x2 + 2x1 = 1

5x1 - 8x2 + 7x3 = 1

OR

Let a1 = (1, -2, -5), a2 = (2, 5, 6) and b = (7, 4, -3) are the vectors. Determine whether b can be generated as a linear combination of a1 and a2. That is determine whether x1 and x2 exist such that x1a1 + x2a2 = b has solution, find it.

8 marks | Asked in 2071

16. Determine if the following system is consistent, if consistent solve the system.

-2x1 - 3x2 + 4x3 = 5

x1 - 2x2 = 4

x1 + 3x2 - x3 = 2

OR

Let   and define a transformation   so that 

a) Find T(u)

b) Find x in R2 whose image under T is b.

8 marks | Asked in 2072

15. Let u and v be nonzero vector in Rn and the angle between them be . Then prove the 


Where the symbol have their usual meanings. 

4 marks | Asked in 2068

16. Given the matrix discuss the for word phase and backward phase of the row reduction algorithm.

8 marks | Asked in 2066

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

solution set. 3x1 + 5x2 - 4x= 0, - 3x1 -2x + 4x3= 0, 6x1 + x2 - 8x3 = 0.

8 marks | Asked in 2067

16. Find a matrix A whose inverse is 

8 marks | Asked in 2069

16. Let   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  and