a. State the properties of inner product for a vector space over C 2

0.

Cc.

Find the LI norm and L2 norm of the vector x =( 1+2j, 2-3], I-j) 3
State and prove Triangular inequality. 4

a. Let x ( 1, 2, 1, 2) and y (2, -3, O, 2). Resolve the vector y into two orthogonal 2

components in wh.ich one is along with x.

b.

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0

௪ ‏له‎

Define orthogonal subspaces of a vector space. What are the important 3 orthogonal
subspaces associated with an mxn matrix A ?

Find the matrix representing the linear transform T : R3 to R3 defined by 4
T(xyz) = (x-y, y-z, z-x) with respect to the basis 8 = { (1,1,0), (0,1,1) , (1,0,1) }

PART C

. Determine the geometric multiplicity and algebraic multiplicity ofthe 63 2

eigenvalues ofthe matrix=O 1 3
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Give Examples for the following 6
1. Hermitian Matrix
. Unitary Matrix

2
3. Orth ogonal Matrix
4

Normal Matrix

. State Spectral Theorem 2

Defi_ne positive definite maheix 2
Let V be an n-dimensional vector space over Cand T: Vto Va linear 8 transform with
distinct eigenvalues {XI,X2, .....,Xk). Prove that the sum of geometric multiplicities of
the eigenvalues is uhnost n.

8 Find the pseudo inverse of the matrix A 2 0

Prove that eigenvectors corresponding to the distinct eigenvalues of a matrix 4 are

linearly independent.