Change of Basis
Now we ask what happens to the matrix of T : V → W when we change bases in V and W. Let {α1, ..., αn} and {β1, ..., βm} be ordered bases for V and W respectively, and suppose we are given a second pair of bases {α'1, ..., α'n} and {β'1, ..., β'm}. Let φ1 and φ2 be the coordinate isomorphisms taking the usual basis in Rn to the first and second bases for V, and let ψ1 and ψ2 be the isomorphisms taking the usual basis in Rm to the first and second bases for W.
Let T1 = ψ1-1 o T o φ1, and T2 = ψ2-1 o T o φ2 (both maps taking Rn to Rm), and let t1 and t2 be their respective matrices. Let p and q be the matrices of the change-of-coordinates automorphisms φ2-1 o φ1 on Rn and ψ2-1 o ψ1 on Rm.
The relationships of these various maps to one another are illustrated in the following commutative diagram.
Since we have T2 = ψ2-1 o T o φ2 = (ψ2-1 o ψ1) o T1 o (φ1-1 o φ2), and since composition of linear maps corresponds to matrix multiplication, it follows that
- t2 = q t1 p-1.
Given that the change of basis has once the basis matrix and once its inverse, this objects are said to be 1-co, 1-contra-variants
Read more about this topic: Change Of Basis, The Matrix of A Linear Transformation
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