Two matrices X, Y ? Mn(R) are said to anti-commute if XY = ?Y X. Let X1, X2, . .
ID: 3403753 • Letter: T
Question
Two matrices X, Y ? Mn(R) are said to anti-commute if XY = ?Y X. Let X1, X2, . . . Xk ? Mn(R) anti-commute (i.e., XiXj = ?XjXi for all i and j) and suppose that X2 i 6= 0n for all i. Show that the matrices X1, X2, . . . Xk are linearly independent..
Two matrices X, Y € M, (R) are said to anti-commute if XY =-YX. Let Xi, X2, . . . X. E M(R) ani-commute (i.e., X,X Mn(R) anti-commute (i.e., Xix, =-X,X, for all i and J) and suppose that X?On n(R) anti-commute (1.e., Ai for all i. Show that the matrices Xi,X2, Xk are linearly independent.Explanation / Answer
suppose there exist scalars c1,c2,c3.......ck
c1y1+c2y2+......+ckyk=0
To show x1,x2,.....xk are linearly dependent we have to show that
c1=c2=c3=.......=ck=0
Given xy=-yx
taking y from this to substitute in above equation gives xy +yx=0
which can be written as y=-2x
substituting this
c1 (-2x1) +c2(-2x2) +........ + ck (-2x2) =0
==> 2(C1x1+C2x2+...........+CkXk) = 0
multiplying both sides by2^-1 yields
2^-1 [ 2(C1x1+C2x2+...........+CkXk) ]= 2^-1*0
2^-1 * 2 is 1
which can be written as
C1x1+C2x2+...........+CkXk=0
This shows x1,x2,x3.....xk are linearly dependent.
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