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linear transformation
This term is a technical cornerstone of linear algebra, describing a specific type of function between vector spaces. It carries a strict mathematical requirement: the function must satisfy the properties of additivity and homogeneity. In a pedagogical context, it is often introduced as the conceptual bridge between abstract vector spaces and concrete matrix multiplication.
While the term "transformation" can be used broadly in geometry to describe any change in position or shape, "linear transformation" specifically excludes translations (shifting a point) unless the space is treated as an affine space. In professional mathematical discourse, it is used with high precision to denote a map that preserves the linear structure of the domain and codomain.
Ý nghĩa
Ví dụ
A linear transformation maps one vector space to another while preserving addition and scalar multiplication.
A linear transformation maps one vector space to another while preserving addition and scalar multiplication.
Can you explain how a matrix represents a linear transformation in three dimensional space?
I need to determine if this specific mapping is actually a linear transformation.
Rotation about the origin is a classic example of a linear transformation.
The professor spent the entire lecture discussing the properties of a linear transformation.
Every linear transformation between finite dimensional vector spaces can be represented by a matrix.
Every linear transformation between finite dimensional vector spaces can be represented by a matrix.
Wait, if the origin is shifted, it is no longer a linear transformation but an affine one.
The kernel of a linear transformation consists of all vectors that map to the zero vector.