Ashwini posted an Question
April 06, 2020 • 19:04 pm 30 points
  • IIT JAM
  • Mathematics (MA)

( let v be the vector space over r consisting of all n x n real matrices for some fixed integer n. which of the following subsets of v are subspaces of v. (a) (

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    Ujjawal vishal best-answer

    option a & b. A subset W of a vector space V is subspace of V , if 1. a , b € W => a+b € W 2. if k € V & a € W => kV € W 3. 0€ W A null matrix is both symmetric and skew as well. We know that sum of two symmetric matrices is a symmetric matrix. Also, if we multiply a symmetric matrix by any scalar we will again get. a symmetric matrix. Hence, both condition meets here Hence, set of symmetric matrices is subspace. Hence the option (a). With similar argument ,option b is also true. Now, if A € U that means A is a nonsingular matrix. Clealry, -A is also nonsingular. but A +(-A) =0 ,so not a nonsingular.Hence , U is not subspace

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