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Check whether each of the following subsets of R3 is linearly independent.
i) f(1;2;3); (
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
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.
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
Find the inverse of the matrix
Let T : R3 !R3 be defined by
T (x1;x2;x3) = (x1
Define T : R3 !R3 by
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
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