A one-to-one linear transformation is a mathematical function that maps each element in one vector space (V) to a unique element in another vector space (W). This means that no two elements in V are mapped to the same element in W.
A one-to-one linear transformation can be represented by a matrix or a set of equations. The matrix representation is a square matrix with n columns and n rows, where n is the dimension of the vector space. The set of equations representation is expressed as a system of linear equations.
If a transformation is not one-to-one, it means that there exists at least one element in V that is mapped to more than one element in W. This can also be referred to as a many-to-one transformation.
A one-to-one linear transformation is invertible, which means that there exists an inverse function that maps each element in W back to a unique element in V. This inverse function allows us to "undo" the original transformation, making it a useful concept in solving equations and finding solutions.
Yes, a linear transformation can be both one-to-one and onto. When this is the case, it is known as an isomorphism. An isomorphism is a bijective linear transformation, meaning that it is both one-to-one and onto, and it preserves the linear structure of the vector spaces.