6.4. Transformations of the Euclidean space, ${ℝ}^3$

So far we have mostly only been considering linear transformations of ${ℝ}^2$, even though the definitions are stated for ${ℝ}^n$. In this section we will briefly consider linear transformations of ${ℝ}^3$, and in particular we would like to understand how to think about them as $3\times 3$ matrices.

As usual, we will consider the standard basis vectors

Analogous to the 2-dimensional case, we define the scalars $a_{ij}$ for $1\le i,j\le 3$ by the equations

and write $A=(a_{ij})$. So $A$ is the matrix associated to the linear transformation $T$.

Conversely, every matrix $A\in M_3(ℝ)$ defines a linear transformation $T:{ℝ}^3\to {ℝ}^3$ by $T(v):=Av$, via matrix multiplication. In this way, we have a bijective correspondence between linear transformations and matrices, as described in Section 6.2.

Example 6.4.1.

• 1. (a)

     Any translation

     $$T:\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \\ \hfill z\hfill \end{array}\right)\mapsto \left(\begin{array}{c}\hfill x+a\hfill \\ \hfill y+b\hfill \\ \hfill z+c\hfill \end{array}\right)\mathit{\hspace{1em}}\text{with}\mathit{\hspace{1em}}\left(\begin{array}{c}\hfill a\hfill \\ \hfill b\hfill \\ \hfill c\hfill \end{array}\right)\ne \left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right)$$

     is not a linear transformation.

  2. (b)

     Let $T:{ℝ}^2\to {ℝ}^2$ be a linear transformation of the plane and $A=(a_{ij})$ its associated $2\times 2$ matrix. Then the map

     $$S:\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \\ \hfill z\hfill \end{array}\right)\mapsto \left(\begin{array}{c}\hfill a_{11}x+a_{12}y\hfill \\ \hfill a_{21}x+a_{22}y\hfill \\ \hfill z\hfill \end{array}\right)=\left(\begin{array}{c}\hfill A\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)\hfill \\ \\ \hfill z\hfill \end{array}\right)$$

     is a linear transformation. Its associated matrix is

     In particular, we can take $T=R_{\theta }$ or $T=H_{\theta }$ as in Proposition 6.2.8 and 6.2.9. They are given by the matrices

  3. (c)

     The reflection in any plane is a linear transformation. For instance, the reflection in the $xy$-plane, which fixes $e_1,e_2$ and sends $e_3$ to $-e_3$ is given by the diagonal matrix .