Linear transformations - enlargement
Linear transformations - enlargement
Understanding Enlargement as a Linear Transformation
- A core idea in linear algebra is linear transformations - operations that alter vectors in certain specific ways.
- An enlargement is a special type of linear transformation that scales all vectors from a common point, known as the centre of enlargement.
- An enlargement can be seen as a scaling transformation, either increasing (magnification) or decreasing (contraction) the lengths of vectors.
The Matrix of an Enlargement
- Linear transformations can be represented by matrices.
- For an enlargement, the matrix is scalar multiplication. The matrix consists of the scaling factor on the principal diagonal (top left to bottom right), with all other entries as zero.
- For example, an enlargement of scale factor 2 has the matrix
[2 0; 0 2].
Investigating Enlargements with Matrix Multiplication
- An enlargement can be carried out by multiplying the matrix representing an object (the coordinates of its vertices as column vectors) by the enlargement matrix.
- Each point of the object is moved along the line from the centre of enlargement, and its distance from the centre of enlargement is multiplied by the scale factor.
Possible Effects of Enlargements
- If the scale factor is greater than 1, an enlargement expands the object.
- If the scale factor is less than 1, an enlargement contracts the object.
- If the scale factor is negative, the enlargement also includes a rotation of 180 degrees around the centre of enlargement.
Notably in Transformations
- It is important to note that an enlargement does not change the shape of an object, just its size and possibly its orientation.
- As a type of linear transformation, enlargements are important in various fields including computer graphics, physics, and engineering.
Remember to practice with plenty of examples to get familiar with how enlargements work in practice and understand their effects on objects.