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AP] ABDUL KALAM TECHNOLOGICAL UNIVERSITY
FIRST SEMESTER M.TECH DEGREE EXAMINATION, DECEMBER 201 8
Branch: ELECTRONICS and COMMUNICATION

Stream:
1) SIGNAL PROCESSING
2) TELECOMMUNICATION ENGINEERING

Course Code & Name: OIEC6301 APPLIED LINEAR ALGEBRA

Answer any twofuU questions from each part
Limit answers to the required points.

Max. Marks: 60 Duration: 3 hours

PART A

1. 2. Define a Field, Give an example for a finite Field 3 0 Show that the set (al,a2, .. .), ai€R,
‏اع‎ ai ) is a vector space. 3 ©. Let V be a vector space and ಟಿ and W are subspaces of ‏.لا‎
‎Define the sum of 3 subspaces U+W. When will be the sum U+W is direct sum ?

Express R3as direct sum of subspaces of R3-

2. a. Find t such that the following vectors are linearly independent. 3 { [ cos(t), sin(t)]T, |-
sin(t), | b Prove that al] bases in a finite dimensional vector space have the same number
3 of vectors.

c. Solve the following linear system of equations using Gauss elimination 3 x+3y-4z 11,
3x-2Y+z -2, X+Y-22 = 1

3. ٤.٦٥۷۸۷ that the subset W of R3given by W — {[x1,xzx3] such that ‏رالا‎ +2, +x3 —O) 5 isa
subspace of 83 . Find a basis of ४४.

b. Let V be an n-dimensional vector space and रि = [സഗ ..... vn) is a basis of 4 V. Show
that every vector in V can be expressed uniquely as a linear combination of the basis
vectors inR.

PART B
4. 2. Let T: 83 to R2is a linear transform defined by T(x,y,z) = (x+z, y+z). 5 Find dimension
of range of T and dimension of null space of T. Verify RankNullity theorem.

0. Let र = 1, -1), (0, 2, -1) and $ = -1, 2), (0, 2, 3) ) be bases 4 of R3.Find R to ऽ change of
basis matrix.