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g:R^4-->R F.E. in 6 var

Source: LIMIT 2020 Cat 2 Obj P2

May 25, 2020
limitfunctionscounting

Problem Statement

The number of functions g:R4→Rg:\mathbb{R}^4\to\mathbb{R} such that, ∀a,b,c,d,e,f∈R\forall a,b,c,d,e,f\in\mathbb{R} :
(i) g(1,0,0,1)=1g(1,0,0,1)=1 (ii) g(ea,b,ec,d)=eg(a,b,c,d)g(ea,b,ec,d)=eg(a,b,c,d) (iii) g(a+e,b,c+f,d)=g(a,b,c,d)+g(e,b,f,d)g(a+e, b, c+f, d)= g(a,b,c,d)+g(e,b,f,d) (iv) g(a,b,c,d)+g(b,a,d,c)=0g(a,b,c,d)+g(b,a,d,c)=0
is :
(A)11 (B)00 (C)infinitely many\text{infinitely many} (D)None of these\text{None of these}
[Hide=Hint(given in question)] Think of matrices
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