Mathematics II 2074

Attempt all questions: 

Group A (10 x 2 = 20)

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

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.

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.

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

5x + 7y = 3

2x + 4y = 1

6. Define vector space with suitable examples.

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}.

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

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

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

Group B (5 x 4 = 20)

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

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. 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 

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.

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

Group C (5 x 8 = 40)

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?

17. Compute the multiplication of partitioned matrices for

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.

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.

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.