# M.A/ M.Sc Mathematics 2nd year for Linear Algebra

**M. A. /M. Sc. – MATHEMATICS, YEAR – 2**** **

**LINER ALGEBRA**** **

### Duration -3 Hours Max Marks: 60

**Note: 1. Attempt any FIVE questions. **

** 2. All questions carry equal marks.**

** **

- a) Show that, f A ? C and B?C then A ? B ? C.

b) For every set A, show that ?UA= A and ??A=?.

- a) Try and prove (A?B)
^{c }=A^{c}UB^{c}

b) The relation R?N*N is defined by (a,b) ? if 5 divides (a-b) is R refleruve symmetric or transitive.

** 3.** a) Prove that properties US7, US9, US10 hold for R”.

b) If u and w are distinct 4 dimensional subspaces of a 6-dimensional vector space v, find the possible dimension of U?W.

** 4**. a) Show that the conditions LT1 and LT2 together imply LT3.

b) Can you show how LT4 and LT5 will follow from LT2.

** 5.** a) Show that the map T:R^{3}?R^{3} defined by T(x_{1},x_{2},x_{3})=(x_{1},x_{2},x_{3})2x_{1}– x_{2},x_{2}+2x_{3})is a linear operator.

b) Show that each basis of U is the dual of some basis of V.

** 6.** a) Reduce the following second degree equation to standard form (Here E R) what is the type of conic they represent.

b) Prove that PF,=ePD where D=dist=once of P from the direction x=a/e Also show that PF_{2}=ePD where D=distance of P from the ex= -a/e.

** 7.** a) which of the following quadratic forms are Orthogonally equivalent?

**a) ** 9x^{2}_{2}+9x^{2}_{3}+12x_{1}x_{2}+12x_{1}x_{3}-6x_{2}x_{3}

**b) **-3y^{2}_{1}+6y^{2}_{2}+6y^{2}_{3}-12y_{1}y_{2}+12y_{1}y_{3}+6y_{2}y_{3}

**c) **11z^{2}-4z^{2}_{2}+11^{2}_{3}+8z_{1}z_{2}=2z_{1}z_{3}+8z_{2}z_{3.}

**d) **Show that the quadratic forms x^{2}-2y^{2}+z^{2} and z^{2}1-2x^{2}1+y^{2}1 are orthogonally equivalent Find the orthogonally transformation which will transform the first of these into the second.

**8**. a) Show that every vector(a,b) ? R^{2} is a linear combination of the vector

(1,0) and (0,1).

b) Prove that IuI IvI cos ?=u.v for any two plane vector u and v,where ? is the angle between them.

**9.** a) Prove that the vector u=(1,2,3,) and v(3,0,-1) are perpendicular.

b) Find the vector equation of the line passing through i+j+k What are the direction cousin of the vector on this line which corresponds it’s the value a=1

Note: Above M.A/ M.Sc Mathematics 2nd year for Linear Algebra has been prepared after reviewing few universities sample papers. We are not sure whether same or similar question papers pattern does your university has or not. We are just trying to help you by providing the possibilities. We always suggest you to go thoroughly with textbooks, teacher’s notes and the syllabus for better preparation of your coming examination.

All the Best!!

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