Change of Basis
Given a linear map whose matrix is A, in the basis B of the space it transforms vectors coordinates as = A. As vectors change with the inverse of B, its inverse transformation is = B.
Substituting this in the first expression
hence
Therefore the matrix in the new basis is A′ = B−1AB, being B the matrix of the given basis.
Therefore linear maps are said to be 1-co 1-contra -variant objects, or type (1, 1) tensors.
Read more about this topic: Linear Map
Famous quotes containing the words change and/or basis:
“All of childhood’s unanswered questions must finally be passed back to the town and answered there. Heroes and bogey men, values and dislikes, are first encountered and labeled in that early environment. In later years they change faces, places and maybe races, tactics, intensities and goals, but beneath those penetrable masks they wear forever the stocking-capped faces of childhood.”
—Maya Angelou (b. 1928)
“That food has always been, and will continue to be, the basis for one of our greater snobbisms does not explain the fact that the attitude toward the food choice of others is becoming more and more heatedly exclusive until it may well turn into one of those forms of bigotry against which gallant little committees are constantly planning campaigns in the cause of justice and decency.”
—Cornelia Otis Skinner (1901–1979)