A linear transformation is a function that takes vectors and maps them to other vectors in the same vector space. This transformation preserves the linear p...
A linear transformation is a function that takes vectors and maps them to other vectors in the same vector space. This transformation preserves the linear properties of the vectors, including the vector addition and scalar multiplication.
Formally, a linear transformation is represented by a linear map, which is a function that satisfies the following properties:
Additivity: T(u + v) = T(u) + T(v)
Homogeneity: T(c * u) = c * T(u)
Examples of linear transformations include:
Scaling: T(u) = cu for any constant c
Projection: T(u) = u onto the subspace perpendicular to the span of u
Rotation: T(u) = u rotated by 90 degrees counterclockwise
Linear transformations can be represented by matrices. A matrix is a rectangular array of numbers that represents a linear transformation. The linear transformation is then applied to the vectors by multiplying the vectors with the matrix.
Linear transformations can be used to solve a variety of problems in mathematics and physics, including finding the eigenvalues and eigenvectors of matrices, solving systems of linear equations, and classifying geometric figures