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A

Section A: Long Answer Questions

Attempt any TWO questions.

3 questions·10 marks each
1long10 marks

Solve the system of linear equations using the Gauss-Jordan method. Discuss the consistency of the system using the rank of the coefficient and augmented matrices.

linear-systemsgauss-jordan
2long10 marks

Define orthogonal and orthonormal sets of vectors. Apply the Gram-Schmidt orthogonalization process to a given set of vectors.

gram-schmidtorthogonal
3long10 marks

Diagonalize the given matrix A by finding a matrix P such that (P^{-1}AP) is diagonal. State the conditions under which a matrix is diagonalizable.

diagonalization
B

Section B: Short Answer Questions

Attempt any EIGHT questions.

9 questions·5 marks each
4short5 marks

Define linearly dependent vectors with an example.

linear-independence
5short5 marks

What is an augmented matrix?

linear-systems
6short5 marks

Define dimension of a vector space.

dimension
7short5 marks

State the conditions for diagonalizability of a matrix.

diagonalization
8short5 marks

Define an inner product space.

inner-product
9short5 marks

Find the eigenvalues of [[4,1],[2,3]].

eigenvalues
10short5 marks

What is the kernel of a linear transformation?

linear-transformation
11short5 marks

Define a Hermitian matrix.

matrix
12short5 marks

State Cramer's rule.

cramers-rule