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.

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.

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

Bornejecbo

Answered 2022-06-27
Author has **19** answers

Regard the vectors in the orthonormal bases $({e}_{1},...,{e}_{n})$ and $({z}_{1},...,{z}_{n})$ 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 $({e}_{1},...,{e}_{n})$ and $({z}_{1},...,{z}_{n})$ 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.

asked 2021-06-13

For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A.

$A=\left[\begin{array}{cccc}2& 3& 5& -9\\ -8& -9& -11& 21\\ 4& -3& -17& 27\end{array}\right]$

Find a nonzero vector in Nul A.

$A=\left[\begin{array}{c}-3\\ 2\\ 0\\ 1\end{array}\right]$

Find a nonzero vector in Nul A.

asked 2021-09-18

Find an explicit description of Nul A by listing vectors that span the null space.

asked 2021-09-13

Assume that A is row equivalent to B. Find bases for Nul A and Col A.

asked 2022-06-24

Let ${T}_{1}$ be a reflection of ${\mathbb{R}}^{3}$ in the xy plane, ${T}_{2}$ is a reflection of ${\mathbb{R}}^{3}$ in the xz plane. What is the standard matrix of transformation ${T}_{2}{T}_{1}$?

Here's my thinking so far:

Since the standard matrix for reflections in xy is

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$

Similarly, standard matrix for orthogonal projection in the xz plane is

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]$

I could multiply

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]\ast \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$

to yield

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

Could someone confirm for me if this is a valid approach?

Here's my thinking so far:

Since the standard matrix for reflections in xy is

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$

Similarly, standard matrix for orthogonal projection in the xz plane is

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]$

I could multiply

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]\ast \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$

to yield

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

Could someone confirm for me if this is a valid approach?

asked 2021-12-16

Is division of matrices possible?

Is it possible to divide a matrix by another? If yes, What will be the result of

asked 2021-03-02

Let T be the linear transformation from R2 to R2 consisting of reflection in the y-axis. Let S be the linear transformation from R2 to R2 consisting of clockwise rotation of

asked 2022-05-22

Linear transformation matrix derivation

$A=\left[\begin{array}{cc}1& 2\\ 0& 3\end{array}\right]\in {\mathbb{R}}^{2\times 2}$

$L:\text{}{\mathbb{R}}^{2\times 2}\u27f6{\mathbb{R}}^{2\times 2};\text{}X\mapsto AX$

Find the transformation matrix with respect to the basis

${\mathcal{B}}_{1}=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\text{}{\mathcal{B}}_{2}=\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right],\text{}{\mathcal{B}}_{3}=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],\text{}{\mathcal{B}}_{4}=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$

$A=\left[\begin{array}{cc}1& 2\\ 0& 3\end{array}\right]\in {\mathbb{R}}^{2\times 2}$

$L:\text{}{\mathbb{R}}^{2\times 2}\u27f6{\mathbb{R}}^{2\times 2};\text{}X\mapsto AX$

Find the transformation matrix with respect to the basis

${\mathcal{B}}_{1}=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\text{}{\mathcal{B}}_{2}=\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right],\text{}{\mathcal{B}}_{3}=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],\text{}{\mathcal{B}}_{4}=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$