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A linear transformation is a mathematical function that maps points from one vector space to another in a linear manner. It preserves the properties of vector addition and scalar multiplication, meaning that the image of a linear combination of vectors is equal to the same combination of their images.
A linear transformation follows the properties of linearity, meaning that the transformation is consistent and predictable. On the other hand, a non-linear transformation does not follow these properties and can result in a more complex and unpredictable output.
In a two-dimensional space, linear transformations can be represented by 2x2 matrices. In a three-dimensional space, they can be represented by 3x3 matrices. These matrices contain the coefficients of the transformation function and can be used to calculate the image of any given point.
Some common examples of linear transformations include rotation, scaling, shearing, and reflection. These transformations are commonly used in computer graphics, physics, and economics to model real-world situations.
Linear transformations can change the size and shape of objects by scaling, rotating, or shearing them. However, these transformations do not alter the basic structure of the object, and the overall shape and size can still be recognized. Non-linear transformations, on the other hand, can significantly alter the shape and size of an object.
