fk378 said:
A linear transformation is a function that maps a vector space to another vector space while preserving the basic algebraic structure of the space. This means that the transformation follows the rules of addition and scalar multiplication, and the origin remains fixed.
A linear transformation can be represented with a matrix by using the coordinates of the input vector as the entries of the matrix, and the resulting vector as the output. The columns of the matrix represent the images of the standard basis vectors in the output space.
To find the matrix of a linear transformation, you first need to choose a basis for the input and output vector spaces. Then, apply the transformation to each basis vector and record the resulting coordinates. These coordinates will form the columns of the matrix, and the matrix will be in the form of [T(v1) T(v2) ... T(vn)] where T is the linear transformation and v1, v2, ..., vn are the basis vectors.
Yes, a linear transformation can have different matrices for different bases. This is because the choice of basis affects the coordinates of the vectors and therefore, the resulting matrix. However, the transformation itself remains the same regardless of the choice of basis.
The properties of a linear transformation, such as injectivity, surjectivity, and dimension of the kernel, can be determined from its matrix by performing row operations and analyzing the resulting matrix. For example, if the matrix is in reduced row echelon form, the number of pivot columns can indicate the dimension of the kernel.