Question

TRUE OR FALSE? If Tis a linear transformation from${\mathit{R}}^{\mathbf{n}\mathbf{}}\mathbf{to}\mathbf{}{\mathit{R}}^{\mathbf{n}}$tosuch that$\mathit{T}\left(\overrightarrow{e_1}\right)\mathbf{,}\mathit{T}\left(\overrightarrow{e_2}\right)\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathit{T}\left(\overrightarrow{e_n}\right)$are all unit vectors, thenmust be an orthogonal transformation.

Short answer

The given statement is false, because the vectors must also be mutually orthogonal.

Step-by-step solution

Recollect the Orthogonal transformations and orthonormal bases theorem.

According to the orthogonal transformations and orthonormal bases theorem, a transformation Tfrom${\mathit{R}}^{\mathbf{n}}$to${\mathit{R}}^{\mathbf{n}}$is orthogonal if and only if the vectors role="math" localid="1661256767897" $\mathit{T}\mathbf{\left(}\overrightarrow{{\mathbf{e}}_{\mathbf{1}}}\mathbf{\right)}\mathbf{,}\mathit{T}\mathbf{\left(}\overrightarrow{{\mathbf{e}}_{\mathbf{2}}}\mathbf{\right)}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathit{T}\mathbf{\left(}\overrightarrow{{\mathbf{e}}_{\mathbf{n}}}\mathbf{\right)}$form an orthonormal basis of ${\mathit{R}}^{\mathbf{n}}$.

Check whether the given statement is a true or false.

The given statement is if T is a linear transformation from $R^n$to$R^n$ such that $T\left(\overrightarrow{{\mathrm{e}}_1}\right),T\left(\overrightarrow{{\mathrm{e}}_2}\right),.....,T\left(\overrightarrow{{\mathrm{e}}_{\mathrm{n}}}\right)$are all unit vectors, thenmust be an orthogonal transformation.

It is given that the vectors$T\left(\overrightarrow{{\mathrm{e}}_1}\right),T\left(\overrightarrow{{\mathrm{e}}_2}\right),.....,T\left(\overrightarrow{{\mathrm{e}}_{\mathrm{n}}}\right)$ are all unit vectors.

But, by the above stated theorem, all the vectors must form an orthonormal basis for $R^n$.

This means, the vectors must also be mutually orthogonal.

So, the given statement is false.