Linear Algebra - Isomorphism, Matrix of Linear Transformation Let T : R^{

vomiderawo 2021-11-12 Answered
Linear Algebra - Isomorphism, Matrix of Linear Transformation
Let T:RnRn be a linear transformation. Let A be the standard matrix for T . Let β be an ordered basis for Rn and B is a matrix whose columns are vectors in β. Prove[T]β=B1AB.
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inenge3y
Answered 2021-11-13 Author has 20 answers

Solution:
Let A be the standard matrix for T where T:RnRn is a linear transformation. Let β be an ordered basis for Rn and B is a matrix whose columns are vectors in β.
Define T(x)=Ax.
Let the vectors of βbev1,v2,vn.
Then,
a1v1+a2v2++anVn=Ax
[v1v2...vn][a1a2an]=Ax
Further we have
B[a1a2an]=Ax(given:B=[v1v2...vn])
[a1a2an]=B1Ax(Right multiply by B1)
Suppose the vector x is any vector from , then x represents any column of B.
Thus, we have[a1a2an]=B1AB
Conclusion
The column matrix [a1a2an] is coordinate vector with respect to B. that is [T]β=[a1a2an]
Hence, it is proved that [T]β=B1AB.

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