Question

1)
a)
Which of the following are binary operations on C? Justify your answer.
i) The operation defined by r y=lwyl.
ii) The operation A defined by x Ay=xy where the bar denotes complex
conjugation.
Also, for those operations which are binary operations, check whether they are
associative and commutative.
(5)
Find the vector equation of the plane determined by the points (1,1,-1),(1,1,1)
and (0,1,1). Also find the point of intersection of the line
r= (1 +31)i + (2-1)j + (1+1)k and the plane.
Check whether the vectors i+j+k, -.1-2j+k are orthonormal.
b)
c)
2)
a)
b)
Check whether the set of vectors v = (1,1,0,1), v2 = (1,0,2,1),
13 = (-1,1, -3, -2) E R* are linearly indpendent. If they are dependent, find aj.
C2 and az ER, not all zero, such that a Vi+ A2V2 + Q3 V3 = 0.
(5)
Which of the following are subspaces of R3? Justify your answer.
i) S= {(x, y, z) € Rx+y = z}
ii) S= {(x,y,z) E R |2x = 3yz}
3)
Let
ple) = {p(x) € R[x]p(x) = p(-x)}
plo) = {p(x) € R[x] p(x) = -p(-x)}​

Answer 1

)

a)

Which of the following are binary operations on C? Justify your answer.

i) The operation defined by r y=lwyl.

ii) The operation A defined by x Ay=xy where the bar denotes complex

conjugation.

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

associative and commutative.

(5)

Find the vector equation of the plane determined by the points (1,1,-1),(1,1,1)

and (0,1,1). Also find the point of intersection of the line

r= (1 +31)i + (2-1)j + (1+1)k and the plane.

Check whether the vectors i+j+k, -.1-2j+k are orthonormal.

b)

c)

2)

a)

b)

Check whether the set of vectors v = (1,1,0,1), v2 = (1,0,2,1),

13 = (-1,1, -3, -2) E R* are linearly indpendent. If they are dependent, find aj.

C2 and az ER, not all zero, such that a Vi+ A2V2 + Q3 V3 = 0.

(5)

Which of the following are subspaces of R3? Justify your answer.

i) S= {(x, y, z) € Rx+y = z}

ii) S= {(x,y,z) E R |2x = 3yz}

3)

Let

ple) = {p(x) € R[x]p(x) = p(-x)}

plo) = {p(x) € R[x] p(x) = -p(-x

DONT KNOW THE ANSWER AOR THE ABOVE QUES