Linear mapping/Dimension formula/No proof/Section
The following statement is called dimension formula.
Theorem
Let denote a field, let and denote -vector spaces, and let
denote a -linear mapping. Suppose that has finite dimension. Then
holds.
Proof
Definition
Let denote a field, let and denote -vector spaces, and let
denote a -linear mapping. Suppose that has finite dimension. Then we call
The dimension formula can also be expressed as
Example
We consider the linear mapping
given by the matrix
To determine the kernel, we have to solve the homogeneous linear system
The solution space is
and this is the kernel of . The kernel has dimension one, therefore the dimension of the image is , due to the dimension formula.
Corollary
Let denote a field, let and denote -vector spaces with the same dimension . Let
denote a linear mapping. Then is injective if and only if is surjective.
Proof
This follows from the dimension formula and fact.