# Need for Separate Basis for Kernel: Explained by Hello

• vish_maths
In summary, the conversation discusses the need for a separate basis for the kernel of a linear mapping, even though the basis for the domain V can also be used to express vectors in the kernel. The reason for this is that the kernel directs the mapping to the zero vector, which is not of interest. A separate basis is needed in order to find a unique representation for vectors in the kernel. The conversation also mentions that while any basis for V spans the kernel, it may not necessarily contain a basis for the kernel itself. However, a basis for the kernel can always be extended to a basis for the entire space.
vish_maths
hello :)

I was trying to prove the following result :
for a linear mapping L: V --> W
dimension of a domain V = dimension of I am (L) + dimension of kernel (L)

So, my doubt actually is that do we really need a separate basis for the kernel ?
Theoretically, the kernel is a subspace of the domain V . So, the basis for V can be used to express any vector in the kernel .

If we really need it, i think this might be the reason ( please check whether i am right or not )

The kernel directs the mapping to the zero vector. We are not interested in this trivial mapping. So, even though the basis vectors of V can clearly describe the vectors in kernel, we keep the vectors in kernel aside and are interested in only those vectors in V which do not produce the zero vector.

Then, since T(ki) = 0 where ki represents any vector in the kernel , ki too ought to be a part of the solution for any mapping. Hence, a separate basis which only can express any vector in the kernel is needed

Am i thinking on the right line ?

Last edited:

Hi,

In what sense do you need to find a basis for the kernel?

vish_maths said:
hello :)

I was trying to prove the following result :
for a linear mapping L: V --> W
dimension of a domain V = dimension of I am (L) + dimension of kernel (L)

So, my doubt actually is that do we really need a separate basis for the kernel ?
Theoretically, the kernel is a subspace of the domain V . So, the basis for V can be used to express any vector in the kernel .

It's true that any basis for V spans the kernel. But the representation would not be unique (unless the kernel is trivial). So a basis for V will be a spanning set, but not a basis, for the kernel.

Ah, I see , that's what he/you meant.

Then Stevel27 is right; unless L==0, the dimension of the space is larger than that

of the kernel, so that a basis for the space contains a linearly-dependent set.

For example, the linear transformation, from R2 to R2, defined by (x, y)-> (x- y, y- x) has {(x, y)| y= x} as kernel. The "standard basis" for R2, {(1, 0), (0, 1)} spans the kernel but does not contain a basis for the kernel. On the other hand, a basis for the kernel can always be extended to a basis for the entire space: {(1, 1)} is a basis for the kernel and any set containing (1, 1) and a vector not a multiple of that, such as {(1, 1), (1, 0)} or {(1, 1), (0, 1)} or {(1, 1), (1, -1)}, is a basis for the entire space.

thank you all :)

## 1. What is the need for a separate basis for kernel?

The kernel is the core of an operating system, responsible for managing the system's resources and providing access to hardware. It is essential for the proper functioning of the system. However, the kernel is also highly sensitive and needs to be protected from other system processes. Thus, a separate basis for the kernel is necessary to ensure its security and stability.

## 2. How does a separate basis for kernel improve system performance?

By isolating the kernel from other processes, a separate basis for the kernel reduces the likelihood of conflicts and interference. This allows the kernel to focus on its primary tasks and improves overall system performance. It also prevents malicious software from accessing the kernel and causing system crashes or slowdowns.

## 3. Is a separate basis for kernel necessary for all operating systems?

Yes, a separate basis for the kernel is necessary for all operating systems, including Windows, Mac, and Linux. While the specific implementation may differ, the concept of isolating the kernel remains the same. This is a fundamental aspect of modern operating system design.

## 4. How does a separate basis for the kernel affect system security?

A separate basis for the kernel greatly enhances system security by limiting access to the core of the operating system. This prevents unauthorized access or modifications to the kernel, which could compromise the entire system. It also allows for more efficient detection and removal of malicious software.

## 5. Are there any drawbacks to having a separate basis for the kernel?

While there are potential drawbacks, such as increased complexity and resource usage, the benefits of a separate basis for the kernel far outweigh them. The added security and improved system performance make it an essential component of modern operating systems.

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