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A

Section A: Long Answer Questions

Attempt any TWO questions.

3 questions·10 marks each
1long10 marks

Define a vector space. State and verify the axioms of a vector space. Show that the set of all 2x2 matrices forms a vector space over the field of real numbers.

vector-spaceaxioms
2long10 marks

Define linear dependence and independence of vectors. Determine whether a given set of vectors is linearly independent. Find a basis and the dimension of the subspace they span.

linear-independencebasis
3long10 marks

Define a linear transformation. Show that a given mapping is a linear transformation and find its matrix representation with respect to standard bases.

linear-transformation
B

Section B: Short Answer Questions

Attempt any EIGHT questions.

9 questions·5 marks each
4short5 marks

Define a vector space with two examples.

vector-space
5short5 marks

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

matrix
6short5 marks

Define a linear transformation.

linear-transformation
7short5 marks

State the rank-nullity theorem.

rank-nullity
8short5 marks

Define an orthonormal set of vectors.

orthogonal
9short5 marks

What is a characteristic equation of a matrix?

eigenvalues
10short5 marks

Define a subspace with an example.

subspace
11short5 marks

Find the trace of the matrix [[1,2,3],[4,5,6],[7,8,9]].

matrix
12short5 marks

State the Cayley-Hamilton theorem.

cayley-hamilton