Matrix Theory/ Isometries

Fix in Consider the real linear map

given by

(a) With respect to the basis B = {1, i, j, k}, find the associated matrix for .
(b) Find the associated matrix A (sub alpha bar) for M (sub alpha bar). Compare with (a).
(c) Compute det(A sub alpha) and det (A sub alpha bar). Interpret.

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...l numbers. We need to make a act on the elements of the basis {1,i,j,k}. We have:
a*1=x+y*i+z*j+t*k
a*i=(x+y*i+z*j+t*k)i=x*i-y-z*k+t*j=-y+x*i+t*j-z*k (we used the fact that ji=-ij=-k and so on).
Similarly:
a*j=-z-t*i+x*j+y*k
a*k=-t+z*i-y*j+x*k
So the matrixA_a is the matrix that has as ...