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A

Section A: Long Answer Questions

Attempt any TWO questions.

3 questions·10 marks each
1long10 marks

Define subspace of a vector space. Show that the intersection of two subspaces is a subspace but the union need not be. Give a counterexample for the union.

vector-spacesubspace
2long10 marks

Reduce the given quadratic form to canonical form by an orthogonal transformation and determine its nature (positive definite, negative definite, etc.).

quadratic-formorthogonal
3long10 marks

Solve the system of linear equations using matrix inversion method and Cramer's rule. Compare the two methods.

linear-systemscramers-rule
B

Section B: Short Answer Questions

Attempt any EIGHT questions.

9 questions·5 marks each
4short5 marks

Define an orthogonal matrix and state its properties.

matrix
5short5 marks

What is meant by the consistency of a system of equations?

linear-systems
6short5 marks

Define the range of a linear transformation.

linear-transformation
7short5 marks

State the Gram-Schmidt orthogonalization process.

gram-schmidt
8short5 marks

Define a quadratic form with an example.

quadratic-form
9short5 marks

Find the eigenvalues of the identity matrix of order 3.

eigenvalues
10short5 marks

Define the nullity of a matrix.

rank-nullity
11short5 marks

What is a unitary matrix?

matrix
12short5 marks

Verify whether (1,1,0), (0,1,1), (1,0,1) span (R^3).

basis