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A

Section A: Long Answer Questions

Attempt any TWO questions.

3 questions·10 marks each
1long10 marks

Define basis and dimension of a vector space. Prove that any two bases of a finite-dimensional vector space have the same number of elements.

basisdimension
2long10 marks

Find the matrix of the linear transformation (T: R^3 \rightarrow R^2) defined by (T(x,y,z) = (x+y, y-z)) with respect to the standard bases, and find its kernel and range.

linear-transformation
3long10 marks

Apply the Gram-Schmidt process to obtain an orthonormal basis from a given basis of (R^3) with the standard inner product.

gram-schmidt
B

Section B: Short Answer Questions

Attempt any EIGHT questions.

9 questions·5 marks each
4short5 marks

Define a basis and dimension with examples.

basis
5short5 marks

Find the rank of a 3x3 identity matrix.

rank
6short5 marks

Define a linear operator.

linear-transformation
7short5 marks

State the spectral theorem for symmetric matrices.

eigenvalues
8short5 marks

What is an orthogonal projection?

orthogonal
9short5 marks

Find the eigenvectors of [[3,0],[0,3]].

eigenvectors
10short5 marks

Define a coordinate vector relative to a basis.

basis
11short5 marks

What is a diagonalizable matrix?

diagonalization
12short5 marks

State two properties of eigenvalues.

eigenvalues