 # Let V be inner product space. Let e 1 </msub> , . . . , e George Bray 2022-06-26 Answered
Let $V$ be inner product space.
Let ${e}_{1},...,{e}_{n}$ an orthonormal basis for $V$
Let ${z}_{1},...,{z}_{n}$ an orthonormal basis for $V$
I have to show that the matrix represents the transformation matrix between ${e}_{1},...,{e}_{n}$ to ${z}_{1},...,{z}_{n}$ is unitary.
You can still ask an expert for help

## Want to know more about Matrix transformations?

• Live experts 24/7
• Questions are typically answered in as fast as 30 minutes
• Personalized clear answers

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it Bornejecbo
Regard the vectors in the orthonormal bases $\left({e}_{1},...,{e}_{n}\right)$ and $\left({z}_{1},...,{z}_{n}\right)$ as column vectors. Then let ${\mathbf{U}}_{e}$ and ${\mathbf{U}}_{z}$ be the matrices where the rows are the transposes of the column vectors ${e}_{1},...,{e}_{n}$ and ${z}_{1},...,{z}_{n}$ respectively. Then both ${\mathbf{U}}_{e}$ and ${\mathbf{U}}_{z}$ are unitary, and the matrix which maps between $\left({e}_{1},...,{e}_{n}\right)$ and $\left({z}_{1},...,{z}_{n}\right)$ is going to be given by ${\mathbf{U}}_{e}{\mathbf{U}}_{z}^{-1}$, which will be unitary because ${\mathbf{U}}_{e}$ and ${\mathbf{U}}_{z}$ are unitary.