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:
“I know what love is. Its understanding. Its you and me and let the rest of the world go by. Just the two of us living our lives together happily and proudly. No self-torture and no doubt. Its enduring and its everlasting. Nothing can change it. Nothing can change us, Ollie. Thats what I think love is.”
—Dewitt Bodeen (19081988)
“This seems to be advanced as the surest basis for our belief in the existence of gods, that there is no race so uncivilized, no one in the world so barbarous that his mind has no inkling of a belief in gods.”
—Marcus Tullius Cicero (10643 B.C.)