A linear mapping, also known as a linear transformation, is a mathematical function that maps one vector space to another in a way that preserves the vector addition and scalar multiplication operations. In simpler terms, it is a function that takes in a vector and outputs another vector while maintaining the same structure and properties.
A linear mapping can be represented by a matrix. The columns of the matrix represent the basis vectors of the input space, while the rows represent the basis vectors of the output space. The entries in the matrix correspond to the coefficients of the linear combination of the basis vectors.
While the terms are often used interchangeably, a linear mapping is a general concept that applies to vector spaces, while a linear function is a specific type of mapping that applies to real numbers. A linear function is a linear mapping between one-dimensional vector spaces.
To prove that a mapping is linear, you must show that it satisfies two conditions: preservation of vector addition and preservation of scalar multiplication. This can be done by applying the properties of linear combinations and using the definition of a linear mapping.
Yes, a linear mapping can have a negative determinant. The determinant of a linear mapping is a measure of how much the mapping distorts space. A negative determinant indicates that the mapping reverses the orientation of the space.