Linear Map - Change of Basis

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:

    Every generation rewrites the past. In easy times history is more or less of an ornamental art, but in times of danger we are driven to the written record by a pressing need to find answers to the riddles of today.... In times of change and danger when there is a quicksand of fear under men’s reasoning, a sense of continuity with generations gone before can stretch like a lifeline across the scary present and get us past that idiot delusion of the exceptional Now that blocks good thinking.
    John Dos Passos (1896–1970)

    The basis of political economy is non-interference. The only safe rule is found in the self-adjusting meter of demand and supply. Do not legislate. Meddle, and you snap the sinews with your sumptuary laws.
    Ralph Waldo Emerson (1803–1882)