# Effect of simple linear transformation

Consider the linear transformation given by

$$T\left\{\begin{bmatrix}x \\y\\z\end{bmatrix}\right\} =\begin{bmatrix}-x\\y\\z\end{bmatrix}$$

Find a matrix $A$ such that $T(x) = Ax$, where x = $[x, y, z]^T$

My attempt: $$\begin{bmatrix}-1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}$$

Describe in simple terms the effect of the linear transformation T on a three-dimensional image.

My attempt: Reflection along the x axis

Is my inteprtretation correct?

• The matrix looks correct. But the interpretation is reflection in the yz plane. The x-axis is normal to this plane. – user_of_math May 26 '16 at 5:53

Let $E=\Bbb{R}^3$ equipped with a frame (O; x, y, z), then we speak about:

A reflection relatively to a plans (mirror) $P$.

Example: the transformation $\Bbb{A=}\left( \begin{array}{ccc} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$indicates a reflection relatively to the plane (o, x, y).

One may ask, for example to find the matrix indicating a reflection in any plane defined by the equation $ax + vy + cz = O$ where $a,b,c$ are constants in $\Bbb{R}$ not all null. (for the simplicity of the mirror , I have imposed $O$ as element of the mirror).

A rotation about an axis $\Delta$.

Example: the transformation $B=\left( \begin{array}{ccc} cos\theta & sin\theta & 0 \\ -sin\theta & cos\theta & 0 \\ 0 & 0 & 1 \end{array} \right)$indicates a rotation about the axis (O; z).

A similar issue is to find a matrix indicating the rotation about any given straight line $\Delta_ {(a, b, c)} = \{\lambda (a, b, c),\; \lambda \in \Bbb{R} \}$ where $(a, b, c)$ is a non zero vector (for the simplicity, I chose the rotation axis contain the origin $O$) .

On inversion relatively to any point $M = (x 0, y_0, z_0)$.

For example, the transformation $C=\left( \begin{array}{ccc} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array} \right)$ indicating an inversion relatively to the origin $O = (0,0,0)$.

Again, a similar question is to find the matrix inversion relatively to any given point $M=(x_0, Y_0, z_0)$ in the space $\Bbb{R}^3$. Thinks