Contraction and Dilation Transformations

Contraction and Dilation Transformation Operators

We will now begin to look at some more interesting aspects of matrices and vectors. One such use arises in linear transformations or linear maps. We will look more into these later on, but for now, we will outline a few common linear transformations before moving onto a more abstract topic of vector spaces.

Definition: For any vector $\vec{x} \in \mathbb{R}^n$ and any scalar $k$ such that $0 ≤ k ≤ 1$, the transformation $T: \mathbb{R}^n \to \mathbb{R}^n$ uniformly contracts all vectors $\vec{x}$ by $k$ towards the origin. If $k ≥ 1$, the transformation $T: \mathbb{R}^n \to \mathbb{R}^n$ uniformly dilates all vectors $\vec{x}$ by $k$ away from the origin. In both cases, $T(x) = kx$

The following images illustrate both a contraction and dilation transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$. Contracted vectors move towards the origin while dilated vectors move away from the origin.

In the example above, we note that for any vector $\vec{x} = (x_1, x_2)$ the following equations represent the image of the contraction/dilation:

(1)

\begin{align} w_1 = kx_1 + 0 x_2 \\ w_2 = 0x_1 + kx_2 \end{align}

We can write this in matrix notation in the following manner:

(2)

\begin{align} \quad \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} = \begin{bmatrix} k & 0\\ 0 & k \end{bmatrix} \begin{bmatrix} x_1\\ x_2 \end{bmatrix} \end{align}

Thus $k$ multiplied by $I$ is the standard matrix for this transformation.