A linear transformation is a function that maps one vector space to another while preserving the operations of vector addition and scalar multiplication. In other words, a linear transformation takes a vector as input and outputs another vector, while maintaining the algebraic structure of the vector space.
Properties of Linear Transformations:
Additivity: T(u + v) = T(u) + T(v) for all vectors u and v.
Homogeneity: T(cu) = cT(u) for all scalars c and vectors u.
Linear transformations are called "linear" because they satisfy these two properties, which are referred to as the "linearity conditions."
Matrix Representation of Linear Transformations:
A linear transformation can be represented as a matrix, which is a rectangular array of numbers. To find the matrix representation of a linear transformation, one can choose a basis for the domain and codomain, and then compute the images of the basis vectors under the transformation.
Kernel and Range:
The kernel of a linear transformation is the set of all vectors in the domain that are mapped to the zero vector in the codomain. The range of a linear transformation is the set of all possible outputs of the transformation.
Invertible Linear Transformations:
A linear transformation is invertible if it has an inverse transformation that maps the codomain back to the domain. Invertible linear transformations are bijective, meaning that they are both injective (one-to-one) and surjective (onto).
Determinant of a Linear Transformation:
The determinant of a linear transformation is a scalar value that reflects the magnitude of the transformation. The determinant can be used to determine if a linear transformation is invertible and to find the inverse transformation.
Eigenvectors and Eigenvalues:
An eigenvector of a linear transformation is a non-zero vector that is mapped to a scalar multiple of itself by the transformation. The scalar is called the eigenvalue associated with the eigenvector. The eigenvalues and eigenvectors of a linear transformation play a key role in understanding the properties and behavior of the transformation.