Question

Consider a linear transformationL from ${\mathbf{R}}^{\mathbf{m}}$to${\mathbf{R}}^{\mathit{n}}$ that preserves length. What can you say about the kernel of L? What is the dimension of the image? What can you say about the relationship between n and m? If Ais the matrix of L, What can you say about the columns of A? What is${\mathbf{A}}^{\mathbf{T}}\mathbf{A}$? What about${\mathbf{AA}}^{\mathbf{T}}$? Illustrate your answer with an example where m=2and n=3.

Short answer

A simple example of such a transformation is $L\left(\overrightarrow{x}\right)=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right]\overrightarrow{x}$,that is $L\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{c}{x}_1\\ x_2\\ 0\end{array}\right]$.

Step-by-step solution

Inequalities between m and n.

According to the theorem 5.3.30

If $L\left(\overrightarrow{x}\right)=\overrightarrow{0}$, then it is written as,

$$\begin{gathered} ||L\left(\overrightarrow{x}\right)||=||\overrightarrow{x}|| \\ =\overrightarrow{0} \end{gathered}$$

So, that it is written as $\overrightarrow{x}=\overrightarrow{0}$.

Therefore, the kernel is $ker\left(L\right)=\left\{\overrightarrow{0}\right\}$.

Now, by theorem 3.3.7,

Consider the terms below,

$$\begin{gathered} dim\left(im\right(L\left)\right)=m-dim\left(ker\right(L\left)\right) \\ =m \end{gathered}$$

Since role="math" localid="1659499215577" ${\mathrm{R}}^{\mathrm{n}}$has an m-dimensional subspace namely $im\left(L\right)$.

The inequality$\mathit{m}\mathbf{\le }\mathit{n}$ holds.

Check AAT is orthogonal or not.

The transformation L preserve right angles (according to theorem 5.3.2).

So that the columns of A are orthonormal.

Since, it is written as $L\left(e{⃗}_1\right),.....,L\left(e{⃗}_m\right)$.

Therefore, it has $A^{T}A=l_m$(proof of theorem 5.3.7).

Since the vectors $\overrightarrow{V_1},....,{\overrightarrow{V}}_m$from an orthonormal basis of im(A), the matrix $A{A}^T$represents the orthogonal projection ontoim(A), by theorem 5.3.10.

Hence, example of a transportation is $L\left(\overrightarrow{x}\right)=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right]\overrightarrow{x}$, that is $L\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{c}{x}_1\\ x_2\\ 0\end{array}\right]$.