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A

Section A: Long Answer Questions

Attempt any TWO questions.

3 questions·10 marks each
1long10 marks

Define inner product space. State and prove the Cauchy-Schwarz inequality.

inner-productinequality
2long10 marks

Find the eigenvalues and eigenvectors of the given matrix and verify the Cayley-Hamilton theorem for it.

eigenvaluescayley-hamilton
3long10 marks

Define kernel and range of a linear transformation. State and verify the rank-nullity theorem with an example.

linear-transformationrank-nullity
B

Section B: Short Answer Questions

Attempt any EIGHT questions.

9 questions·5 marks each
4short5 marks

Define the row space and column space of a matrix.

vector-space
5short5 marks

Solve x + y = 3, 2x - y = 0 using matrices.

linear-systems
6short5 marks

Define the span of a set of vectors.

vector-space
7short5 marks

State the Cauchy-Schwarz inequality.

inner-product
8short5 marks

What is the geometric multiplicity of an eigenvalue?

eigenvalues
9short5 marks

Define a positive definite matrix.

quadratic-form
10short5 marks

What is the standard basis of (R^3)?

basis
11short5 marks

Define an idempotent matrix.

matrix
12short5 marks

Find the norm of the vector (3, 4) in (R^2).

inner-product