Question

1. Recall that given a basis, the space of linear endomorphisms of R", End (R"), can be identified with the space of nxn matrices. Let us denote this space by Mat (n). Clearly, with respect to standard addition of matrices and multiplication by scalars, Mat (n) is a na-dimensional vector space. 1. Let X e Mat (n). Then, we can think as being coordinates on Mat (n). 1,j=1...n Clearly, we must have (1 if i = k and j = 1 1 0 otherwise Express (1) in terms of Kronecker deltas. 2. Consider the product functio. p: Mat (n) * Mat (n) - Mat (n) - ven by p(X,Y) = XY (2) Recall that (XY); = x¿Y. By taking {X}},m=1..n and {Ym}1,m=1...n as coordinates on Mat (n) Mat (n), work out the partial derivatives 1 and 1 and conclude that p is a differentiable function on Mat (n) x Mat (n). 8p? Om

Please write neatly and legibly. Please show all work.

Answer 1

* Solution : Given that Recall that given a basis, the space of Linear endomorphism of eEnd (R"), can be identified wilt the Space of nxn Mataices. I The Multiplication by Scolars - Matin) ig an 'n?-dimentimal vector space SO) (1) Let xe Matin), then we can think of {x}]ij=i-609 being co-ordinates on Matin) clearly, we must have 8 % (1 it ick and jaz Sxí / o ; other wise Since we have two conditions, we must have two Kroneckers dellas. They fxK P: Mat (n) x maten) → Mat(m) () consider the product Function P(X,Y)=XY + (x,y); = té via Note that p = xkyk Till

Símillaolly X clearlly are continuous for 1sij,k,esny thus 'D' ġi a differentiable Function on Matin) x Malin)