Mathematics II 2068

Attempt all questions:

Group A (10 x 2 = 20)

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

2. What is meant by independent of vectors?

3. What is normal form of a matrix?

4. Define nonsingular linear transformation with suitable example.

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

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

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

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

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

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

Group B (5 x 4 = 20)

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).

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.

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)

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).

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. 

Group C(5 x 8 = 40)

16. Test for consistency and solve:

2x - 3y + 7z = 5

3x + y - 3z = 13

2x + 19y - 47z = 32

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.

18. Verify Cayley-Hamilton Theorem for matrix:

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).