CSIR NET Linear Algebra Most Probable Question Series | CSIR NET Mathematical Science 2024 | IFAS

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[@dikshapatle1331]

The correct answer is (B) The image of T is infinite dimensional. The kernel of T is trivial, since T(x) = 0 implies x = 0. So, option (A) is false. The image of T is the set of all sequences of the form (x + x1, x1 + x2, ...), which includes all sequences with equal adjacent terms. This is an infinite-dimensional subspace of RN, since it includes sequences like (1, 1, 0, 0, ...), (0, 1, 1, 0, ...), etc. The quotient vector space RN/T(RN) is isomorphic to the kernel of T, which is trivial, so it is finite-dimensional. Therefore, only option (B) is correct.

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[@Naminami-w1f]

LA -option b will be correct

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