Intro
In linear algebra and its applications, the concept of kernels is very useful, providing us with the solutions to many problems we encounter.
Finding the kernel can be time-consuming for complex problems, which explains the great excitement arising after Jack Dongarra developed a groundbreaking method to effectively approximate kernels.
In 2021, Jack J. Dongarra was the recipient of the equivalence to the Nobel Prize in computer science, the Turing Award. The jury's motivation was:
For his pioneering contributions to numerical algorithms and libraries that enable high performance computational software to keep pace with exponential hardware improvements for over four decades
With huge contributions to the BLAS and LAPACK, there is no doubt that Dongarra is a hero in the field.
Concept
In the context of linear algebra, the term image does not refer to pictures as we are used to. Instead, image is short for image space, which simply is another name for the range of the linear transformation. In other words, the image of is the set of all possible outputs resulting from multiplying any vector by .
In linear algebra, image of refers to the range of the linear transformation
A property of all matrix multiplications is that . Consequently, is always in the image. However, may not be the only vector giving rise to as the output.
The set of all vectors producing when multiplied by is called the kernel of , and is denoted by
Math
Per definition, the kernel of matrix is the set of vectors satisfying:
As previously discussed in the course, this equation forms a homogeneous system of linear equations, where we now use the term kernel to refer to its solution set.
Similarly, the image of is nothing but another term for the set of vectors that are solutions to the equation
The multiplication of matrix can be viewed as a linear transformation and the terms kernel and image only make sense in this context.
Kernel
The kernel of a linear transformation refers to all vectors that maps to the zero vector and is often noted as ker. An alternative explanation is that the kernel is the solution set to the homogeneous system of equations . The kernel is also usually called the null space, with the two terms being virtually synonymous. However, if you were to ask a mathematician, there is a high risk that they would counter that the kernel and the null space are not equivalent concepts, but are essentially the same concept because they share the same definition. The semantic difference is that the kernel is meant for a linear transformation, while the null space is meant for the standard matrix for a linear transformation. In practical terms, in a basic course in linear algebra, both expressions are usually treated as equivalents.
If we return to the definition of the kernel, we can algebraically refer to the following definition.
Let:
be a linear transformation with the domain and the codomain . Then we call the set of all vectors that satisfy the following equation to :
as the kernel, which can be described as a set of vectors in relation to the linear transformation as:
The visual definition of the kernel is in the following image, noted as ker:
The kernel of three transformations
Let us take the three linear transformations of projection, reflection and rotation as examples and list their kernels:
Projection - the set of all vectors that are orthogonal to the object to which the projection refers
Reflection - zero vector only
Rotation - zero vector only
Image
The image of a linear transformation refers to all vectors in the codomain that maps from at least one vector in the domain. An alternate definition is that the image of a linear transformation refers to the set of all of its possible transformations. A third formulation may be that the image of a linear transformation is all possible linear combinations of its standard matrix columns.
It is not the intention to confuse the definition of the image with three different examples. The purpose is to offer more formulations, so that the chance that one of them can be perceived as understandable to the beginner is higher. Before the exam, however, all three definition examples should be perceived as understandable.
The image and the range are unequivocally equivalent expressions, while the column space is usually regarded practically as an equivalent expression. However, a mathematician can argue that the column space is essentially the same concept as image / range because the definitions are analogous, but the semantic difference is that the image / range is meant for a linear transformation, while the column space is meant for the standard matrix of a linear transformation. In practical terms, in a basic course in linear algebra, all three expressions are usually treated as equivalents.
The definition of the image / range can be algebraically defined as:
Let:
be a linear mapping with domain and codomain . Then we call the set of all vectors that is a solution to the equation:
the image, transformation, and range of . It can also be described as a set of vectors in relation to the linear transformation as:
The visual definition of the image can be found in the following image, noted as Im:
The image for three transformations
Let us take the three linear transformations projection, reflection and rotation as examples and list their images:
Projection - the subspace which is the object to which the projection refers.
Reflection - all space
Rotation - all space
