Chapter 5, Problem 1E

If T is a linear transformation from ${ℝ}^n$ to ${ℝ}^n$ suchthat $T\left({\stackrel{\to }{e}}_1\right),T\left({\stackrel{\to }{e}}_2\right),\mathrm{...},T\left({\stackrel{\to }{e}}_n\right)$ are all unit vectors, thenT must be an orthogonal transformation.

To determine

True or False:T must be an orthogonal transformation.

Answer to Problem 1E

True.

Explanation of Solution

Given information:T is a linear transformation from ${ℝ}^n$ to ${ℝ}^n$ such that $T\left(\overrightarrow{e_1}\right),T\left(\overrightarrow{e_2}\right),T\left(\overrightarrow{e_3}\right),\mathrm{...},T\left(\overrightarrow{e_n}\right)$ are all unit vectors.

Definition used:A linear transformation T from ${ℝ}^n$ to ${ℝ}^n$ is called orthogonal if it preserves the length of the vectors: $\Vert T\left(\overrightarrow{x}\right)\Vert =\Vert \overrightarrow{x}\Vert$ for all $\overrightarrow{x}$ in ${ℝ}^n$

Since $\Vert T\left(\overrightarrow{e_i}\right)\Vert =\overrightarrow{e_i}$ for $i=1,2,3,\mathrm{..}n$ , the transformation preserves the length of the vectors.

  $\Rightarrow$ T must be an orthogonal transformation.

Hence, it is true that T must be an orthogonal transformation.