# IAS Mains Mathematics 2020 Expected Questions For 2020 Exam

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Which of the following sets of vectors α= (a

_{1}, a2, a3…a_{n}) in R^{n}are subspaces of R^{n}(n≥3)?All α such that a

_{1}≤0;All α such that a

_{3}is an integer;All α such that a

_{2}+4a_{3}=0;All α such that a

_{2}is rational.Let C be the field of complex numbers and let n be a positive integer (n≥2).Let V be the vector space of all n×n matrices over C. Which of the following sets of matrices A in V are subspaces of V?

All invertible A;

All non-invertible A;

All A such that AB=BA, where B is some fixed matrix in V.

Prove that the union of two subspaces is a subspace if and only if one is contained in the other.

Ans:

Therefore X+Y belong to X. then it means (X+Y)-X belong to X. i.e. Y belong to X. that is Y contained in X.

Let V be the vector space of all functions from R into R; let V

_{e}be the subset of even functions , f(–x)=f(x); let V_{o}be the subset of odd functions, f(–x)=–f(x)Prove that V

_{e}and V_{o}are subspaces of V.Prove that V

_{e}+V_{o}=V.Prove that V

_{e}∩ V_{0}= {0}.In V

_{3}(R),where R is the field of real numbers, examine the following sets of vectors for linear independence:{(1,3,2),(1,-7,-8),(2,1,-1)};

{(1,2,0),(0,3,1),(-1,0,1)}.

Find a linearly independent subset T of the set S={α

_{1,}α_{2,}α_{3,}α_{4}} where α_{1}=(1,2,-1),α_{2}=(-3,-6,3), α_{3}=(2,1,3),α_{4}=(8,7,7)єR^{3}which spans the same space as S.Find whether the vectors 2x

^{3}+x^{2}+x+1, x^{3}+3x^{2}+x-2, x^{3}+2x^{2}-x+3 of R[x], the vector space of all polynomials over the real number field , are linearly independent or not.Determine whether or not the following vectors form a basis of R

^{3}: (1,1,2),(1,2,5),(5,3,4).Show that the vectors α

_{1}=(1,0,-1),α_{2}=(1,2,1),α_{3}=(0,3,-2) form a basis of R^{3}.Express each of the standard basis vectors as a linear combination of α_{1},α_{2}and α_{3}.Show that the set S={1,x,x

^{2},……,x^{n}} of n+1 polynomials in x is a basis of the vector space P_{n}(R), of all polynomials in x (of degree at most n) over the field of real numbers.“Corresponding to each subspace W

_{1}of a finite dimensional vector space V(F),there exists a subspace W_{2}such that V is the direct sum of W_{1}and W_{2}.” Prove the theorem.Let V be the vector space of all polynomial functions of degree less than or equal to two from the field of real numbers R into itself. For a fixed tєR, let g1(x) =1, g

_{2}(x) =x+t, g_{3}(x) =(x+t)^{2}.Prove that {g

_{1}, g2, g3} is a basis for V and obtain the coordinates of c_{0}+c_{1}x+c_{2}x^{2}.Show that the mapping T:V

_{2}(R)―›V_{3}(R) defined as T(a,b)=(a+b,a-b,b) is a linear transformation from V_{2}(R) into V_{3}(R).Find the range,rank,null space and nullity of T.Let T: R3―›R

^{3}be the linear transformation defined by: T(x, y, z) =(x+2y-z, y+z, x+y-2z). Find a basis and the dimension of (i) the range of T; (ii) the null space of T.Describe explicitly a linear transformation from V

_{3}(R) into V_{3}(R) which has its range spanned by (1, 0,-1) and (1, 2, 2).Let T be a linear operator on V

_{3}(R) defined by T (a,b,c)=(3a,a-b, 2a+b+c) for all (a,b,c)єV_{3}(R).Is T invertible? If so, find a rule for T^{-1}like the one which defines T.Let V(R) be the vector space of all polynomials in x with coefficients in R of the form f(x) =a

_{0}x^{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}i.e. the space of polynomials of degree three or less. The differential operator D is a linear transformation on V. The set B= {α_{1}… α4} where α_{1}=x^{0}, α2=x^{1}, α3=x^{2}, α4=x^{3}is an ordered basis of V. Write the matrix of D relative to the ordered basis B.Let T be the linear operator on R

^{3}defined by T(a,b,c)=(3a+c, -2a+b, -a+2b+4c).What is the matrix of T in the standard ordered basis B for R

^{3}?Find the transition matrix p from the ordered basis B to the ordered basis B’={α

_{1},α_{2},α_{3}}, where α_{1}=(1,0,1), α_{2}=(-1,2,1),and α_{3}=(2,1,1).Hence find the matrix of T relative to the ordered basis B’.Find all (complex) characteristic values and characteristic vectors of the following matrices:

1 1 1 (b) 1 1 1

1 1 1 0 1 1

1 1 1 0 0

Let T be the linear operator on R

^{3}which is represented in the standard basis by the matrix

-9 4 4

-8 3 4

-16 8 7

Prove that T is diagonalizable.

Show that similar matrices have the same minimal polynomial.

Show that every square matrix is

*uniquely*expressible as the sum of a symmetric and a skew-symmetric matrix.If A is a square matrix of order n, prove that | Adj (Adj A) | =|A|

^{(n-1)ᴧ2}Find the rank of the matrix

2 -2 0 6

4 2 0 2

A= 1 -1 0 3

1 -2 1 2 by reducing it to normal form.

Discuss for all values of k the system of equations

2x+3ky+ (3k+4) z=0,

x+ (k+4) y+ (4k+2) z=0,

x+2(k+1) y+ (3k+4) z=0.

Investigate for what values of α and µ the simultaneous equations

x+y+z=6, x+2y+3z=10, x+2y+αz=µ

Have (i) no solution, (ii) a unique solution, (iii) an infinite number of solutions.

Show that the three equations

-2x+y+z=a, x-2y+z=b, x+y-2z=c

Have no solutions unless a+b+c=0, in which case they have infinitely many solutions. Find these solutions when a=1, b=1, c= -2.

Show that if a diagonal matrix is commutative with every matrix of the same order, then it is necessarily a scalar matrix.

Find the possible square roots of the two rowed unit matrix I.

Show that the matrix B’AB is symmetric or skew-symmetric according as A is symmetric or skew-symmetric.