charlies1902 said:
A linear transformation is a mathematical function that maps one vector space to another while preserving the operations of addition and scalar multiplication. In simpler terms, it is a function that takes in a vector and outputs another vector, where the output is a linear combination of the input vectors.
A linear transformation can be represented using a matrix. The columns of the matrix represent the images of the basis vectors of the input vector space. The transformation can then be applied by multiplying the input vector with the matrix.
A linear transformation preserves the structure and properties of vector spaces, while a nonlinear transformation does not. This means that a linear transformation follows the rules of addition and scalar multiplication, while a nonlinear transformation does not.
A transformation can be considered linear if it satisfies two properties: additivity and homogeneity. Additivity means that the transformation of the sum of two vectors is equal to the sum of the individual transformations. Homogeneity means that the transformation of a scalar multiple of a vector is equal to the scalar multiple of the transformation of the original vector.
No, a linear transformation cannot change the dimension of a vector space. The dimension of the output vector space will always be equal to or less than the dimension of the input vector space. This is because a linear transformation can only map vectors within the same vector space, and the dimension of a vector space is determined by the number of basis vectors.
