Finding the inverse of linear transformation using matrix. A linear transformation represented by a matrix with respect to some random bases, how could I find the inverse of the transformation using the matrix representation?

Hugo Stokes 2022-10-18 Answered
Finding the inverse of linear transformation using matrix. A linear transformation represented by a matrix with respect to some random bases, how could I find the inverse of the transformation using the matrix representation?
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Answers (1)

getrdone07tl
Answered 2022-10-19 Author has 23 answers
Let T be an invertible linear transformation from an n-dimensional vector space to another, A its ( n × n ) matrix, B = { α 1 , , α n } a basis for the domain, B = { β 1 , , β n } a basis for the co-domain, and β = y 1 β 1 + + y n β n a vector in the co-domain.
The inverse of T , T 1 ,, is related to A 1 by [ T 1 β ] B = A 1 [ β ] B ,, where [ ] B denotes the coordinate matrix relative to the ordered basis B . Then with A i j 1 representing the i j entry of A 1 ,, we compute
[ T 1 β ] B = A 1 [ β ] B = A 1 [ y 1 β 1 + + y n β n ] B = A 1 [ y 1 y n ] = [ i = 1 n A 1 i 1 y i i = 1 n A n i 1 y i ] ; T 1 β = α 1 i = 1 n A 1 i 1 y i + + α n i = 1 n A n i 1 y i .
If you are satisfied having your inverse in terms of B ,, we are done.
If, however, you want the inverse in terms of the standard ordered basis, express each α i in terms of the standard basis, and group the terms in the last expression above by the elements of the standard basis.
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