dylanhouse said:
A linear transformation is a mathematical function that maps a vector from one vector space to another. It preserves the basic operations of addition and scalar multiplication, and also maintains the straightness of lines and planes.
A linear transformation follows the properties of linearity, such as preserving addition and scalar multiplication, while a non-linear transformation does not. Non-linear transformations can also change the shape of geometric objects, while linear transformations maintain their shape.
A linear transformation can be represented by a matrix, where the columns of the matrix represent the images of the basis vectors of the original vector space. It can also be represented using a system of linear equations.
Linear transformations are widely used in many fields, such as physics, engineering, and economics. They allow for the simplification and efficient representation of complex systems and data, making them essential in modeling and solving real-world problems.
A transformation is considered linear if it follows the properties of linearity, such as preserving addition and scalar multiplication. This can be tested by applying the transformation to a linear combination of vectors and checking if it satisfies the definition of linearity. Another method is to represent the transformation using a matrix and checking if it follows the rules of matrix multiplication.
