Tessellate S.T.E.M.S (2019) - Mathematics - Category C - Set 2 - Objective Problem 2

Geometry Level 4

Let S S be the real vector space of all symmetric 2018 × 2018 2018\times 2018 matrices with real entries which have trace 0 0 , under the usual addition and scaling by reals. Find dim ⁡ R S \dim_{\mathbb{R}}S .

2037171 2037170 2035153 2035152

Tessellate S.T.E.M.S. Mathematics by Tessellate S.T.E.M.S. Ma… , 26

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1 solution

Patrick Corn
Jan 18, 2019

Let n = 2018. n = 2018. A basis for S S consists of:

(1) The ( n − 1 ) n / 2 (n-1)n/2 symmetric matrices consisting of all zeroes plus a symmetric pair of off-diagonal 1 1 s

(2) The n − 1 n-1 diagonal matrices consisting of a 1 1 in the top left corner and a − 1 -1 somewhere else on the diagonal.

So the dimension of S S is n − 1 + ( n − 1 ) n / 2 = ( n − 1 ) ( n + 2 ) / 2 = 2017 ⋅ 1010 = 2037170 . n-1 + (n-1)n/2 = (n-1)(n+2)/2 = 2017 \cdot 1010 = \fbox{2037170}.

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