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Check whether each of the following subsets of R3 is linearly independent.

i) f(1;2;3); (

i) f(1;2;3); (

Let

P(e)(x) = fp(x) 2 R[x]jp(x) = p(

P(e)(x) = fp(x) 2 R[x]jp(x) = p(

i) The operation 5 defined by x5y = jln(xy)j where ln x is the natural

logarithm.

ii) The operation 4 defined by x4y = x2+y3.

Also, for those operations which are binary operations, check whether they are

associative and commutative

logarithm.

ii) The operation 4 defined by x4y = x2+y3.

Also, for those operations which are binary operations, check whether they are

associative and commutative

Find the radius and the center of the circular section of the sphere jrj = 26 cut off

by the plane r (2i+6j+3k) = 70.

by the plane r (2i+6j+3k) = 70.

Reduce the conic x2+6xy+y2

Which of the following statements are true and which are false? Justify your answer with

a short proof or a counterexample.

i) R2 has infinitely many non-zero, proper vector subspaces.

ii) If T : V !W is a one-one linear transformation between two finite dimensional

vector spaces V andW then T is invertible.

iii) If Ak = 0 for a square matrix A, then all the eigenvalues of A are zero.

iv) Every unitary operator is invertible.

v) Every system of homogeneous linear equations has a non-zero solution

a short proof or a counterexample.

i) R2 has infinitely many non-zero, proper vector subspaces.

ii) If T : V !W is a one-one linear transformation between two finite dimensional

vector spaces V andW then T is invertible.

iii) If Ak = 0 for a square matrix A, then all the eigenvalues of A are zero.

iv) Every unitary operator is invertible.

v) Every system of homogeneous linear equations has a non-zero solution

Find the inverse of the matrix

2

4

2

4

Let T : R3 !R3 be defined by

T (x1;x2;x3) = (x1

T (x1;x2;x3) = (x1

Define T : R3 !R3 by

T(x;y;x) = (x+y;y;2x

T(x;y;x) = (x+y;y;2x

Check whether the following system of equations has a solution.

4x+2y+8z+6w = 3

2x+2y+2z+2w = 1

x+3z+2w = 3

4x+2y+8z+6w = 3

2x+2y+2z+2w = 1

x+3z+2w = 3