# IAS Mains Mathematics 2017 Expected Questions

1. Which of the following sets of vectors α= (a1, a2, a3…an) in Rn are subspaces of Rn(n≥3)?

• All α such that a1≤0;

• All α such that a3 is an integer;

• All α such that a2+4a3=0;

• All α such that a2 is rational.

2. 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.

3. 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.

4. Let V be the vector space of all functions from R into R; let Ve be the subset of even functions , f(–x)=f(x); let Vo be the subset of odd functions, f(–x)=–f(x)

• Prove that Ve and Vo are subspaces of V.

• Prove that Ve+Vo=V.

• Prove that Ve ∩ V0= {0}.

5. In V3(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)}.

6. Find a linearly independent subset T of the set S={α1,α2,α3,α4} where α1=(1,2,-1),α2=(-3,-6,3), α3=(2,1,3),α4=(8,7,7)єR3 which spans the same space as S.

1. Find whether the vectors 2x3+x2+x+1, x3+3x2+x-2, x3+2x2-x+3 of R[x], the vector space of all polynomials over the real number field , are linearly independent or not.

2. Determine whether or not the following vectors form a basis of R3: (1,1,2),(1,2,5),(5,3,4).

3. Show that the vectors α1=(1,0,-1),α2=(1,2,1),α3=(0,3,-2) form a basis of R3.Express each of the standard basis vectors as a linear combination of α12 and α3.

1. Show that the set S={1,x,x2,……,xn} of n+1 polynomials in x is a basis of the vector space Pn(R), of all polynomials in x (of degree at most n) over the field of real numbers.

2. “Corresponding to each subspace W1 of a finite dimensional vector space V(F),there exists a subspace W2 such that V is the direct sum of W1 and W2.” Prove the theorem.

3. 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, g2(x) =x+t, g3(x) =(x+t) 2.

4. Prove that {g1, g2, g3} is a basis for V and obtain the coordinates of c0+c1x+c2x2.

5. Show that the mapping T:V2(R)―›V3(R) defined as T(a,b)=(a+b,a-b,b) is a linear transformation from V2(R) into V3(R).Find the range,rank,null space and nullity of T.

1. Let T: R3―›R3 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.

2. Describe explicitly a linear transformation from V3(R) into V3(R) which has its range spanned by (1, 0,-1) and (1, 2, 2).

3. Let T be a linear operator on V3(R) defined by T (a,b,c)=(3a,a-b, 2a+b+c) for all (a,b,c)єV3(R).Is T invertible? If so, find a rule for T-1 like the one which defines T.

4. Let V(R) be the vector space of all polynomials in x with coefficients in R of the form f(x) =a0x0+a1x+a2x2+a3x3 i.e. the space of polynomials of degree three or less. The differential operator D is a linear transformation on V. The set B= {α1… α4} where α1=x0, α2=x1, α3=x2, α4=x3 is an ordered basis of V. Write the matrix of D relative to the ordered basis B.

5. Let T be the linear operator on R3 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 R3?

• Find the transition matrix p from the ordered basis B to the ordered basis B’={α123}, where α1=(1,0,1), α2=(-1,2,1),and α3=(2,1,1).Hence find the matrix of T relative to the ordered basis B’.

6. 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

7. Let T be the linear operator on R3 which is represented in the standard basis by the matrix

-9 4 4

-8 3 4

-16 8 7

Prove that T is diagonalizable.

8. Show that similar matrices have the same minimal polynomial.

9. Show that every square matrix is uniquely expressible as the sum of a symmetric and a skew-symmetric matrix.

10. If A is a square matrix of order n, prove that | Adj (Adj A) | =|A|(n-1)ᴧ2

11. 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.

1. 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.

2. 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.

3. 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.

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

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

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

## Discussions & Questions

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