IAS Mains Mathematics 2017 Expected Questions

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 noninvertible 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}+x2, 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,ab,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+2yz, y+z, x+y2z). 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,ab, 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 skewsymmetric matrix.

If A is a square matrix of order n, prove that  Adj (Adj A)  =A^{(n1)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, x2y+z=b, x+y2z=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 skewsymmetric according as A is symmetric or skewsymmetric.