![]() $$To summarise, a "transformation matrix in the same basis" and a "change of basis matrix" are just two specific cases of that general concept of a matrix of a linear transformation with respect to two specific bases. That said, a particular vector #X# can be changed (its components can be changed while its direction and magnitude remain the same) from a basis #A# to a different basis #B# using a change-of-basis matrix #M_# be the matrix of #I# with respect to #A# and #B#,$$ Each basis expresses the components as two different numbers. There are infinite possible bases to choose from. ![]() Any vector in the 2D space can be expressed as a linear combination of the two basis vectors in the chosen basis. A basis is a set of two independent (unit or not) vectors. Let's consider a vector #X# in 2D with its two components #(x_1, x_2)_A# expressed in the basis #A#.
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