Linear Transformations: An In-Depth Guide for Advanced University-Level Notes

Linear Transformations:

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:

  1. Additivity: T(u + v) = T(u) + T(v) for all vectors u and v.

  2. 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.

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