# [[Dilation Transformation]] (Mathematics > Linear Algebra) ## Definition A **dilation transformation** is a linear transformation that scales all vectors by a constant factor $k$ with respect to the origin. It multiplies the standard basis vectors by $k$, and is represented by the matrix: $ \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}. $ ## Key Concepts - **Scale Factor ($k$):** Determines how much each vector is stretched or contracted. - **Linear Transformation:** A mapping that preserves vector addition and scalar multiplication. - **Matrix Representation:** Dilation can be represented by a diagonal matrix with $k$ on the diagonal. - **Geometric Scaling:** Dilation scales distances, areas, and volumes by factors of $|k|$, $k^2$, and $k^n$ respectively. ## Important Properties 1. **Uniform Scaling:** Every vector is scaled by the same factor $k$, preserving the direction (if $k>0$) or reversing it (if $k<0$). 2. **Area/Volume Scaling:** In $\mathbb{R}^2$, areas are multiplied by $k^2$, and in $\mathbb{R}^3$, volumes are multiplied by $k^3$. 3. **Determinant:** The determinant of the dilation matrix is $k^2$, which indicates the factor by which areas (or volumes) are scaled. ## Essential Formulas - **Transformation Formula:** $ T(\mathbf{v}) = k\mathbf{v}. $ - **Matrix Representation:** $ T = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}. $ ## Core Examples 1. **Basic Example:** For a dilation with scale factor $k=2$, the transformation matrix is: $ T = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}. $ This doubles the length of any vector in $\mathbb{R}^2$. 2. **Advanced Application:** Applying $T$ to the unit square spanned by the standard basis $\{\mathbf{i}, \mathbf{j}\}$, the square is expanded to a new square with side length 2 and area $4$. ## Related Theorems/Rules - **Similarity Transformations:** Dilation is a type of similarity transformation that preserves the shape of objects. - **Matrix Multiplication:** Dilation matrices can be composed with other linear transformations via standard matrix multiplication. ## Common Pitfalls - **Orientation Misunderstanding:** A positive scale factor preserves orientation, while a negative scale factor reverses it. - **Overlooking Area/Volume Effects:** Failing to square or cube the scale factor when calculating the new area or volume. - **Mixing Transformations:** Confusing dilation with other types of linear transformations such as rotations or shears. ## Related Topics - [[Linear Transformation]] - [[Matrix Multiplication]] - [[Scalar Multiplication of Matrices]] - [[Determinants]] - [[Matrix-Vector Multiplication]] - [[Standard Matrix of a Linear Transformation]] - [[Rigid Motions]] - [[Rotation by 90°, 180°, 270°]] - [[Horizontal Scaling of the Sine Function]] - [[Horizontal Scaling of the Tangent Function]] - [[Transformed Sine and Cosine Functions]] - [[Transformed Secant and Cosecant Functions]] - [[Area of a Parallelogram Using Determinants]] - [[Area Under a Curve]] - [[Area Bounded by Curves]] - [[Area_by_Integrating_with_Respect_to_y]] - [[Cross Product and Parallelogram Area]] - [[Scalar Triple Product Volume]] - **Rotation Transformations:** Alter the direction of vectors while preserving lengths. - **Shear Transformations:** Distort shapes without uniform scaling. - **Reflection Transformations:** Flip objects over a line or plane. ## Quick Review Questions 1. How does a dilation transformation with scale factor $k$ affect the length of a vector in $\mathbb{R}^2$? 2. What is the determinant of the dilation matrix, and what does it represent in terms of area scaling? *** ![[Dilation_Transformation_visualization]]