Mathematics II - Unit Wise Questions

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

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]

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

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

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

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

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.

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

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

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

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

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

one solution.

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

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

2. Define singular and nonsingular matrices.

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. Define linear transformation between two vector spaces.

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

2. What is meant by independent of vectors?

2. Write numerical importance of partitioning matrices.

2. When is a linear transformation invertible?

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

3. Is  invertible matrix?

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

3. Solve the system 

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

by using the inverse of the matrix 

3. Show that the matrix is not invertible.

3. What is normal form of a matrix?

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

3. Define invertible matrix transformation. 

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.

4. Define invertible matrix transformation.

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 

4. Define invertible linear transformation.

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

4. Define nonsingular linear transformation with suitable example.

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

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

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

5. Show that the matrices do not commute.

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?

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

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

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

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

5x + 7y = 3

2x + 4y = 1

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 

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

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 

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

6. Define vector space with suitable examples.

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

6. Define vector space.

6. Define vector space.

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

6. Define subspace of a vector space.

6. When is a linear transformation invertible.

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

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

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}.

7. Determine if  is a Nul(A) for 

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

7. Define subspace of a vector with an example.

5. Change into reduce echelon form of the matrix 

7. Find the rank of AB where 

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

6. Determine if  is Nul(A), where, 

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

8. Is  an Eigen value of  ?

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

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

8. Are the vectors;  eigen vectors of  

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

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

8. Show that 7 is an eigen value of 

8. Is an Eigen value of  

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

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

9. If S = {u₁,... .... ... ... , u_p} 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.

9. Define kernel and image of linear transformation.

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

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

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

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

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

two vectors in  Rⁿ. 

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

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

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

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

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

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

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

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

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

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

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

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

11. Determine if the given set is linearly dependent:

11. If a set s = {v₁, v_{2,  ... ... ... ,}v_p} in Rⁿ contains the zero vector, then prove that the set is linearly dependent. Determine if the set  is linearly dependent.

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

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

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

11. Let  and  . Find the images under T of 

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

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

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.

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.

12. Find the determinant of 

12. If  compute (Ax)^T, x^TA^T and xx^T. Can you compute x^TA^T? 

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

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.

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

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

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.

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

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).

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 

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.

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.

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

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

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

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.

14. Find the eigen values of  

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].

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

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.

14. Find the eigen values of .

14. Find the eigen values of 

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)

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).

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

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.

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)

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.

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.

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

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

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).

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.

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 define Then

a) Find T(u)

b) Find an 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.

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.

16. Let a₁ = (1, 2, -5), a₂ = (2,5,-3) and b = (7,4, -3). Determine whether b can be generated as a linear combination of a₁ and a₂. That is, determine whether x₁ and x₂ exists such that x₁a₁ + x₂a₂ = b

has the solution , find it.

OR

Determine if the following system is consistent

x₂ - 4x₃ = 8

2x₁ - 3x₂ + 2x₃ = 1

5x₁ - 8x₂ + 7x₃ = 1

16. Find a matrix A whose inverse is 

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. 

16. Let  be a linear transformation and let A be the standard matrix for T. Then prove that: T map Rⁿ on to R^m if and only if the columns of A span R^m; 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?

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

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

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

17. Find the inverse of  if it exists, by using elementary row reduce the augmented matrix.

17. Test the consistency and solve

x + y + z = 4

x + 2y + 2z = 2

2x + 2y + z = 5

OR

Verify Cayley Hamilton theorem for matrix 

17. Compute the multiplication of partitioned matrices for 

17. Compute the multiplication of partitioned matrices for 

17. Compute the multiplication of partitioned matrices for

17. If the consumption matrix C is

and the final demand is 50 units for manufacturing, 30 units for agriculture and 20 units for services, find the production level x that will satisfy this demand.

OR

Compute the multiplication of partitioned matrices for

17. If the consumption matrix C is 

and the final demand is 50 units for manufacturing, 30 units for agriculture and 20 units for services, find the production level x that will satisfy this demand.

17. An n x n matrix A is invertible if and only if A is row equivalent to I_n , and in this case, any sequence of elementary row operations that reduces A to I_n also transform I_{n x m} into A^-1. Use this statement to find the inverse of the matrix if exist.

18. What do you mean by change of basis in Rⁿ? Let and consider the bases for R² given by B = {b₁, b₂} and C = {c₁, c₂}.

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?

18. Let be a basis for a vector space V. Then the coordinate mapping is one-to-one linear transformation from V into Rⁿ. Let 

a) Show that the set  is a basis of R³.

b) Find the change of coordinates matrix for to the standard basis.

c)  Write the equation that relates x in R³ to . 

d) Find for the x give above.

18. Let v₁ = (3, 6, 2), v₂ = (-1, 0, 1), x = (3, 12, 7) and B = {v₁, v₂}. Then B is a basis for H = span {v₁, v₂}. Determine if x is in H, and if it is, find the co-ordinate vector of x relative to B. 

18. Let b₁ = (1, 0, 0), b₂ = (-3, 4, 0), b₃ = (3, -6, 3) and x = (-8, 2, 3) then 

(a) Show that B = {b₁, b₂, b₃} is a basis of R³.

(b) Find the change of co-ordinates matrix from B to the standard basis.

(c) Find [X]_B, for the given x.

18. Let b₁ = (1,0,3), b₂ = (1,-1,2) and x = (3,-5,4). Does B={b1, b2, b3} form a basis? Find [x]_B, for x.

18. The set of matrices of the form 

is a subspace of the vector 3 x 3 matrices. Verify it.

16. Test for consistency and solve:

2x - 3y + 7z = 5

3x + y - 3z = 13

2x + 19y - 47z = 32

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  i.e., x = 3b₁ + b₂. Find [x]_c.

OR

Define basis of a subspace of a vector space.Let  where   and let  Show that span {v₁, v₂, v₃} = span {v₁, v₂}  and find a basis for the subspace H.

18. What do you mean by change of basis in Rⁿ ? Let 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 if possible.

19. Diagonalize the matrix if possible.

OR

Suppose A = PDP^-1, where D is a diagonal n x n matrix. If  is the basis for Rⁿ formed for the columns of P, then prove that D is the matrix for the transformation. Define , where Find a basis for R² with the property that the matrix for T is a diagonal matrix.

19. Diagonalize the matrix, if possible

19. Diagonalize the matrix, if possible 

19. Diagonalize the matrix  if possible 

OR

Find the eigen value of   and find a basis for each eigen space.

17. Let U and V be vector spaces over a field and assume that dim U=dim V. If   is a linear transformation, then prove that the following are equivalent;

i. T is invertable

ii. T is one-one and onto, and 

iii. T is non-singular 

OR 

Verify that the set of matrices of the form  is a subspace of the vector space of 3 x 3 matrices.

19. Diagonalize the matrix  if possible.

19. Diagonalize the matrix, if possible 

20. Find the equation  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?

18. Verify Cayley-Hamilton Theorem for matrix:

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 A

20. What is a least squares solution? Find a least squares solution of Ax = b, where 

OR

What do you understand by orthonormal set? Show that {v₁, v₂, v₃} is an  orthonormal basis of R³, where

Prove that an m x n matrix U has orthonormal columns if and only if U^TU = 1.

20. What is a least-squares solution? Find a least-squares solution of Ax = b, where

20. Find a least-square solution for Ax = b with  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 A^TAx = A^Tb.

20. Find the equation y = a₀ + a₁x for the least squares line that best fits the data points (2, 1), (5, 2), (7, 3),(8, 3).

OR

When two vectors 4 and v are orthogonal? If u and vectors, prove that [dist (u, -v)]² = [dist (u, v)]² if u.v = 0.

19. Let V and W be the vector spaces over a field F of real numbers. Let dim V = n and dim W = m. Let {e₁,e₂,... ...,e_n} be a basis of V and {f₁, f₂, ... ... ... , f_m} be a basis of W. Then, prove that each linear transformation can be represented by an m x n matrix A with elements from F such that 

Y = AX

Are column matrices of coordinates of relative to its basis and coordinates of relative to its basis respectively.

OR

Compute the multiplication partitioned matrices for 

20. When two vectors u and v orthogonal? If u and v are vectors, prove that  [dist(u, -v)]² = [dist(u, v)]² if u, v = 0.

OR

Find a least square solution of Ax = b for 

20. Find the equation for the least squares line that best fits the data points(2, 1),(5, 2),(7,3),(8, 3).

19. Diagonalize the matrix 

OR

Compute the multiplication of partitioned matrices for

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).

2. Define linear transformation with an example.        [1+1+3+5]

    Let , , , 

    and define a transformation T: R²→R² and T(x) = Ax then

    (a) find T(v)

    (b) find x ∈ R² whose image under T is b.

2. Define linear transformation with an example. Check the following transformation is linear or not?  be defined by T(x, y) = (x, 2y).Also, let T( x,y)= ( 3x+y, 5x+7y, x+3y). Show that T is a one- to-one linear transformation. Does T maps onto ?

4. Let T is a linear transformation. Find the standard matrix of T such that

(i) by T(e1) = (3, 1, 3, 1) and T(e₂) = (-5, 2, 0, 0) where e₁ = (1, 0) and e₂ = (0, 1); 

(ii) rotates point as the origin through radians counter clockwise.

(iii) Is a vertical shear transformation that maps e₁ into e₁-2e₂ but leaves vector e₂ unchanged.

6. Let , and define  be defined by

T( x) = Ax, find the image under T of  and .

7. Let   and define T: R² →R² by T(x) = Ax, find the image under T of

         and 

6. Define linear transformation with an example. Is a transformation defined by T(x, y) = (3x + y, 5x + 7y, x + 3y) linear? Justify.

6. Let us define a linear transformation. Find the image under   

2. What is the condition of a matrix to have an inverse? Find the inverse of the matrix If it exists.

3. Find the LU factorization of

3. Find the LU factorization of

2. What is the condition of a matrix to have an inverse? Find the inverse of the matrix in exists.

5. Compute u+ v, u-2v and 2u+v where .