Just one more thing, if a transformation is injective then it is not necessarily onto?
I ask this because I would like to know if in determining whether a transformation is invertible, if it is sufficient to look at whether or not it is injective.
A linear transformation is a mathematical function that maps one vector space to another, while preserving the operations of vector addition and scalar multiplication. In simpler terms, it is a way to transform one set of vectors into another set of vectors using a set of rules.
The kernel of a linear transformation, also known as the null space, is the set of all vectors in the domain that get mapped to the zero vector in the codomain. In other words, it is the set of all vectors that the linear transformation "ignores" or maps to the origin.
To find the kernel of a linear transformation, you can set up a system of equations using the transformation's matrix representation and solve for the variables. The solutions to the system of equations will form the basis for the kernel.
The kernel and range of a linear transformation are complementary subspaces. This means that the dimensions of the kernel and range add up to the total dimension of the vector space. Additionally, any vector in the range can be written as the sum of a vector in the kernel and a vector in the range.
The kernel of a linear transformation is important because it allows us to understand the behavior of the transformation. It helps us determine if the transformation is one-to-one (injective) or onto (surjective), and it provides a way to find solutions to systems of linear equations. Additionally, the kernel is useful in applications such as image and signal processing, where it can help identify patterns and reduce noise.