Mathematics II - Unit Wise 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]

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

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

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

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

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

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

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.

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. Write numerical importance of partitioning matrices.

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 linear transformation between two vector spaces.

2. What is meant by independent of vectors?

2. When is a linear transformation invertible?

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

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

3. Show that the matrix is not invertible.

3. Solve the system 

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

by using the inverse of the matrix 

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

3. Is  invertible matrix?

3. What is normal form of a matrix?

3. Define invertible matrix transformation. 

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

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

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

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

4. Define invertible matrix transformation.

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

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. 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. Show that the matrices do not commute.

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

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

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

5x + 7y = 3

2x + 4y = 1

6. Define vector space.

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

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

6. Define vector space with suitable examples.

6. Define vector space.

5. Change into reduce echelon form of the matrix 

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. Find the rank of AB 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. Determine if w = (1, 3, -4) is a Nul A, where 

7. Define subspace of a vector with an example.

7. Determine if  is a Nul(A) for 

6. Determine if  is Nul(A), where, 

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

8. Show that 7 is an eigen value of 

8. Is an Eigen value of  

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

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

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

8. Is  an Eigen value of  ?

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

two vectors in  Rⁿ. 

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

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

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

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

9. Define kernel and image of linear transformation.

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

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

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

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

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

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

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

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

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

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

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. Determine if the given set is linearly dependent:

11. Let  and  . Find the images under T of 

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

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

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

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

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

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

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. 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.

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

12. Find the determinant of 

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.

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

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

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.

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

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

14. Find the eigen values of  

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

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

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 

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

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

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

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 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.

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).

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. 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 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. 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.

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

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. Find a matrix A whose inverse is 

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. 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

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. 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.

17. Compute the multiplication of partitioned matrices for 

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. 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.

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

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. 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. 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 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. 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 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. 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.

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.

19. Diagonalize the matrix, if possible 

19. Diagonalize the matrix, if possible 

19. Diagonalize the matrix  if possible.

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.

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 

OR

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

19. Diagonalize the matrix, if possible

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. 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. 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?

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. 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 

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. 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 .

7. Let and . What value(s) of k, if any, will make AB = BA?

7. Let Determine the value (s) of k if any will make AB = BA.

7. Let What value (s) of k if any will make AB = BA?

12. Find the QR factorization of the matrix 

12. Find LU factorization of the matrix 

8. Compute det A, where .

8. Define determinant. Evaluate without expanding 

8. Define determinant. Compute the determinant without expanding 

3. Define linearly independent set of vectors with an example. Show that the vectors (1, 4, 3), (0, 3, 1) and (3, -5, 4) are linearly independent. Do they form a basis? Justify.

6. When two column vectors in R² are equal? Give an example. Compute u+3v, u-2v where                [1+4]

    , 

5. For what value of h will y be in span {v₁ , v₂, v₃} if 

9. Define null space of a matrix A. Let    

         , and 

    Then show that v is in the null A.

10. Verify that 1^k, (-2)^k, 3^k are linearly independent signals.

9. Define null space . Find the basis for the null space of the matrix 

9. Define subspace of a vector space. Let Show that H is a subspace of:

10. Let B = {b₁, b₂} and C = (c₁, c₂) be bases for a vector  V, and suppose b₁ = -c₁ + 4c₂ and b₂ = 5c₁ - 3c₂. Find the change of coordinate matrix for a vector space and find [x]_c for x = 5b₁ + 3b₂.

10. Find the dimension of the null space and column space of 

15. Find the vector x determined by the coordinate vector where

15. State and prove the Pythagorean theorem of two vectors and verify this for u = (1, -1) and v = (1, 1).

9. Let H be the set of all vectors of the form . Show that H is a subspace of .

10. Find basis and the dimension of the subspace

    .

11. If . Find a formula Aⁿ, where A = PDP^-1 and 

             and 

8. Find the eigen values of

11. Find the eigenvalues and eigenvectors of .

11. Find the eigenvalues of the matrix 

11. Find the eigen values of the matrix 

4. Find a least square solution of the inconsistent system Ax= b for

4. Find a least square solution of the inconsistent system Ax = b for

    , 

3. Find the least-square solution of Ax=b for 

4. Find the least-square solution of Ax = b for 

12. Define orthogonal set. Show that {u₁, u₂, u₃} is an orthogonal set, where

u₁ = , , 

12. Find a unit vector v of u = (1, -2, 2,3) in the direction of u.

13. Prove that the two vectors u and v are perpendicular to each other if and only if the line through u is perpendicular bisector of the line segment from -u to v.

13. Let , where  and  . Construct an orthogonal basis {v₁, v₂} for W.

14. Let * be defined on  by . Then show that  forms a group.

14. Let an operation * be defined on Q⁺ by a*b = ab/2. Then show that Q⁺ forms a group.

13. Define group. Show that the set of all integers Z forms group under addition operation.

13. Define binary operation. Determine whether the binary operation * is associative or commutative or both where * is defined on Q by letting . 

15. Define ring and show that set of real numbers with respect to addition and multiplication operation is a ring.        [2+3]

15. Define ring with an example. Compute the product in the given ring (12)(16) in .

14. Define ring with an example. Compute the product in the given ring (-3, 5) (2, -4) in Z₄ x Z₁₁.

14. Show that the ring is an integral domain.