Let

vomiderawo
2021-11-12
Answered

Linear Algebra - Isomorphism, Matrix of Linear Transformation

Let$T:{R}^{n}\to {R}^{n}$ be a linear transformation. Let A be the standard matrix for T . Let $\beta$ be an ordered basis for $R}^{n$ and B is a matrix whose columns are vectors in $\beta$ . Prove${\left[T\right]}_{\beta}={B}^{-1}AB$ .

Let

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inenge3y

Answered 2021-11-13
Author has **20** answers

Solution:

Let A be the standard matrix for T where

Define T(x)=Ax.

Let the vectors of

Then,

Further we have

Suppose the vector x is any vector from , then x represents any column of B.

Thus, we have

Conclusion

The column matrix

Hence, it is proved that

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