What is Kernel: Definition and 211 Discussions

The Linux kernel is a free and open-source, monolithic, modular, multitasking, Unix-like operating system kernel. It was conceived and created in 1991 by Linus Torvalds for his i386-based PC, and it was soon adopted as the kernel for the GNU operating system, which was created as a free replacement for UNIX. Since then, it has spawned a plethora of operating system distributions, commonly also called Linux.
Linux is deployed on a wide variety of computing systems, such as embedded devices, mobile devices (including its use in the Android operating system), personal computers, servers, mainframes, and supercomputers. It can be tailored for specific architectures and for several usage scenarios using a family of simple commands (that is, without the need of manually editing its source code before compilation); privileged users can also fine-tune kernel parameters at runtime. Most of the Linux kernel code is written using the GNU extensions of GCC to the standard C programming language and with the use of architecture specific instructions (ISA). This produces a highly optimized executable (vmlinux) with respect to utilization of memory space and task execution times.Day-to-day development discussions take place on the Linux kernel mailing list (LKML). Changes are tracked using the version control system git, which was created by Torvalds as a bespoke replacement for BitKeeper. Linux as a whole is released under the GNU General Public License version 2 (GPLv2), but it also contains several files under other compatible licenses, and an ad hoc exemption for the user space API header files (UAPI).

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  1. S

    MHB Kernel of integral eq of positive terms only

    Again it's the Fredholm integral equation of the 2nd type, that is $y(x)=f(x)+\int_{a}^{b} \,k(x,t) y(t) dt$ , where $a,b>0$ I am taking the norm of the integral operator equal to $||K||= max ([a,b] \int_{a}^{b} \,|k(x,t)| dt$ the kernel is represented by a series of positive terms only or...
  2. T

    What are the image and kernel of matrices A and B?

    Homework Statement I know this stuff isn't complicated but the definitions of my book are very formal and confusing. I have to find the image and the kernel of these two matrices: A= \begin{bmatrix}1&2&3\end{bmatrix} B= \begin{bmatrix}2&3\\6&9\end{bmatrix} The Attempt at a Solution my book...
  3. C

    Please help. What is the relation between the kernel of A an

    Homework Statement What is the relation between the kernel of A and the kernel of (A^2 + A)? Homework EquationsThe Attempt at a Solution Break into A^2x = 0 and Ax = 0. We know Ax = 0 because that's the kernel of A, ker(A^2x) is subset of ker(A) so ker(A^2 + A) is a subset of ker (A)?
  4. A

    Kernel vector of statics Jacobian

    Hi all, I was reading an article that utilized a 3x4 statics Jacobian and said to calculate the kernel vector: You can row by row, where Where Ai is the statics Jacobian with the ith column removed. The problem is I have a 3x3 statics Jacobian, so if I remove the ith column I will end up...
  5. Paul Shredder

    Structure of a Matrix With Empty Null Space

    Hi guys, I hope you are having a great day, this is Paul and, as you have seen in the title, that's what I'm looking for, let me explain: When you have a square matrix with empty null space, that is, the only solution to the equation Ax=0 (with dim(A)=n x n) is the vector x=0n x 1, means that...
  6. PsychonautQQ

    How can the kernel of a ring morphism be a subring?

    I don't understand this page, https://www.proofwiki.org/wiki/Kernel_of_Ring_Homomorphism_is_Subring, but how can this be a true statement? Wouldn't a ring morphism map the multiplicitive identity to itself? So it wouldn't be in the kernel, so how could the kernel be a subring? I happened upon...
  7. M

    Finding Kernel, Image, Rank, Nullity of Matrix

    Homework Statement Find Kernel, Image, Rank and Nullity of the matrix 1 −1 1 1  | 1 2 −1 1 | 0 3 -2 0 Homework EquationsThe Attempt at a Solution I have reduced the matrix into rref of 3 0 1 3 0 3-2 0...
  8. P

    How to find matrix with a given image or kernel rather than vice versa?

    I'm interested in learning how to solve a relatively general sort of problem that comes up a lot in my problem sets and will presumably come up in future exams. I'm asked to give an example of a matrix or linear transformation that has a given image or kernel. Here are some examples...
  9. TrickyDicky

    Possible title: When Does the Kernel of a Homomorphism Reduce to the Identity?

    I'm centering on lie group homomorphisms that are also covering maps from the universal covering group. So that if their kernel was just the identity they would be isomorphisms. Are there situations in which the kernel of such a homomorphism would reduce to the identity? I'm thinking of...
  10. M

    Dual basis and kernel intersection

    The problem statement, all variable Let ##\phi_1,...,\phi_n \in V^*## all different from the zero functional. Prove that ##\{\phi_1,...,\phi_n\}## is basis of ##V^*## if and only if ##\bigcap_{i=1}^n Nu(\phi_i)={0}##. The attempt at a solution. For ##→##: Let ##\{v_1,...,v_n\}## be...
  11. W

    Linear Algebra: Kernel, Basis, Dimensions, injection, surjections

    Homework Statement The Attempt at a Solution Can someone please check my work?
  12. V

    Kernel, Basis, Rank: Hints & Answers

    Please see attached question In my opinion this question is conceptional and abstract.. For part a and b, I think dim(Ker(D)) = 1 and Rank(D) = n but I do not know how to explain them For part c What I can think of is if we differentiate f(x) by n+1 times then we will get 0 Can...
  13. M

    Finding Kernel of P: Steps to Show Ker(P) is in Ker(P°P)

    I have a linear transformation P:z→z I want to show the Kernel (p) is a subset of the kernel (P ° P) I know that the composite function is defined by (P ° P)(x)=P(P(x)) Where do I begin with this? To find ker(P) I would do P(x)=0 but I am not sure how I would do this here. What steps...
  14. M

    Proving Ker P ∈ Eigenspace for Eigenvalue 0 in Linear Transformation

    Lets say you have a linear transformation P. The eigenvalues of the matrices are 0,1 and 2. How would you show that ker P belongs to the eigenspace corresponding to 0? So you have an eigenvalue 0. Let A be the 3X3 matrix. I was thinking of doing something like Ax=λx and substitute 0 for λ...
  15. B

    Write the subspace spanned by vectors as a kernel of a matrix.

    Hi Lets say I have a vectorspace in Rn, that is called V. V = span{v1,v2,... vk} Is it then possible to create an m*n matrix A, whose kernel is V. That is Ax = 0, x is a sollution if and only if x is an element of V. Also if this is possible, I imagine that k may not b equal to m?
  16. J

    Integral equation with bounded unknown kernel

    I need to solve an integral equation of the form $$\forall \omega \in [0,1], ~ \int_{\mathbb{R}} K(\omega,y)f(y)dy = \omega$$ where - f is known and positive with $$\int_{\mathbb{R}} f(y)dy = 1$$ - K: [0,1] x R -> [0,1] is the unknown kernel I am looking for a solution other than...
  17. J

    MHB Finding the Image and Kernel

    Consider d map f:R^4 into R^2 defines by f(x,y,z,w)=(2x+y+z+w,x+z-w). find the image and the kernel, please include explanations...
  18. D

    Solving an integration equation with unknown kernel

    I am pondering over how to solve the following (seemingly nonstandard) integral equation. Let h(t) be a known function which is non-negative, strictly increasing and satisfies that h(t) → 0 as t→-∞ and h(t)→1 as t→∞. Indeed, h(t) can be viewed as a cumulative distribution function for a...
  19. B

    Surjection between kernel and image of a homomorphism

    Hi, I was wondering whether the following is true at all. The first isomorphism theorem gives us a relation between a group, the kernel, and image of a homomorphism acting on the group. Could this possibly also imply that there exists a surjective homomorphism either mapping the previous kernel...
  20. G

    Find the smallest Matrix for this Kernel

    Hey, i have a question, i know the Kernel, but i have to find the smallest Matrix for this Kernel, how can i do that ? Thank you!
  21. L

    Existence of Linear Operators with Matching Subspaces in Vector Spaces

    Homework Statement Prove for every subspace B of vector space C, there is at least 1 linear operator L: C→C with ker (L) = B and there's at least 1 linear operator L':C→C with L'(C) = B. Homework Equations The Attempt at a Solution The first operator with Ker(L) = B would be...
  22. S

    Kernel and Range of a Linear Mapping

    Homework Statement Find the kernel and range of the following linear mapping. b) The mapping T from P^{R} to P^{R}_{2} defined by T(p(x)) = p(2) + p(1)x + p(0)x^{2} The Attempt at a Solution I'm not sure how to go about this one. Normally I would use the formula T(x) = A * v...
  23. F

    Linear algebra: eigenvalues, kernel

    Homework Statement I've tried to solve the following exercise, but I don't have the solutions and I'm a bit uncertain about result. Could someone please tell if it's correct? Given the endomorphism ##\phi## in ##\mathbb{E}^4## such that: ##\phi(x,y,z,t)=(x+y+t,x+2y,z,x+z+2t)## find: A) ##...
  24. stripes

    Uniform convergence for heat kernel on unit circle

    Homework Statement I would like to use the Weierstrass M-test to show that this family of functions/kernels is uniformly convergent for a seminar I must give tomorrow. H_{t} (x) = \sum ^{-\infty}_{\infty} e^{-4 \pi ^{2} n^{2} t} e^{2 \pi i n x} . Homework Equations The Attempt at a...
  25. A

    When is the kernel of a linear operator closed?

    If you consider a bounded linear operator between two Hausdorff topological vector spaces, isn't the kernel *always* closed? I mean, if you assume singleton sets are closed, then the set \{0\} in the image is closed, so that means T^{-1}(\{0\}) is closed, right (since T is assumed continuous)? I...
  26. D

    Kernel subsets of transformations

    Homework Statement Let T_1,T_2:ℝ^n\rightarrowℝ^n be linear transformations. Show that \exists S:ℝ^n\rightarrowℝ^n s.t. T_1=S\circ T_2 \Longleftrightarrow kerT_2\subset kerT_1 . The Attempt at a Solution (\Longrightarrow) Let S:ℝ^n\rightarrowℝ^n be a linear transformation s.t...
  27. M

    Exploring the Impact of Poisson Kernel on Math Functions

    Why Poisson kernel is significant in mathematics? Poisson kernel is ##P_r(\theta)=\frac{1-r^2}{1-2rcos\theta+r^2}##. http://www.math.umn.edu/~olver/pd_/gf.pdf page 218, picture 6.15. If we have some function for example ##e^x,sinx,cosx## what we get if we multiply that function with Poisson...
  28. T

    Linear transformation, subspace and kernel

    Hi We have a linear transformation g : ℝ^2x2 → ℝ g has U as kernel, U: the 2x2 symmetric matrices (ab) (bc) A basis for U is (10)(01)(00) (01)(10)(01)I thought this would be easy but I've been sitting with the problem for a while and I have no clue on how to solve it...
  29. D

    MHB Poisson kernel significance

    What is the significance of the Poisson kernel (besides solving the Dirichlet problem)? What is the Poisson's role in solving the Dirichlet problem? I know it is the solution but what is meant by its role?
  30. N

    Find the Kernel of the Trace of a Matrix

    Homework Statement Let F : Mnn(R) → R where F(A) =tr(A). Show that F is a linear transformation. Find the kernel of F as well as its dimension. What is the image of F? Homework Equations The Attempt at a Solution I have shown that it is a linear transformation. But I am not...
  31. D

    MHB What is the Limit of the Poisson Kernel Prove for $r\to 1$?

    Prove: $$ \lim_{r\to 1}P(r,\theta) = \begin{cases} \infty, & \theta = 0\\ 0, & \text{otherwise} \end{cases} $$ For the first piece, take the summation $$ P(1,0) = \frac{1}{\pi}\left(\frac{1}{2} + \sum_{n = 1}^{\infty} 1^n\right). $$ Then $\sum\limits_{n = 1}^{\infty} 1^n = \infty$. Therefore, we...
  32. D

    MHB Prove Evenness of Poisson Kernel for Fixed $r$

    For a fixed $r$ with $0\leq r < 1$, prove that $P(r,\theta)$ is an even function.Take $-r$. Then \begin{alignat*}{3} P(-r,\theta) & = & \frac{1}{2\pi}\frac{1 - (-r)^2}{1 - 2(-r)\cos\theta + (-r)^2}\\ & = & \frac{1}{2\pi}\frac{1 - r^2}{1 + 2r\cos\theta + r^2} \end{alignat*} I have $1 +...
  33. D

    MHB Is Poisson's Kernel Useful for Computing Sums of Cosine Functions?

    $$ P(r,\theta) = \frac{1}{2\pi}\sum_{n = -\infty}^{\infty}r^{|n|}e^{in\theta} \overbrace{=}^{\mbox{?}} \frac{1}{\pi}\left[\frac{1}{2} + \sum_{n=1}^{\infty}r^n\cos n\theta\right] $$ Is this true?
  34. matqkks

    Kernel and Range: Understanding Linear Transformation in Algebra

    Why are we interested in looking at the kernel and range (image) of a linear transformation on a linear algebra course?
  35. V

    Solving for the Unknown Integral Kernel?

    Consider, f(\mathbf{w}) = \int K(\mathbf{w,\mathbf{v}}) g(\mathbf{v}) d\mathbf{v} where \mathbf{v},\mathbf{w} \in \mathbb{R^3}. Is it possible to solve for the integral kernel, K(\mathbf{w,\mathbf{v}}) , if f(\mathbf{w}) and g(\mathbf{v}) , are known scalar functions and we require...
  36. R

    Gaussian integers, ring homomorphism and kernel

    Homework Statement let \varphi:\mathbb{Z}[i]\rightarrow \mathbb{Z}_{2} be the map for which \varphi(a+bi)=[a+b]_{2} a)verify that \varphi is a ring homomorphism and determine its kernel b) find a Gaussian integer z=a+bi s.t ker\varphi=(a+bi) c)show that ker\varphi is maximal ideal in...
  37. F

    Mathematica Mathematica remote kernel hangs after password

    Hey all, I'm trying to setup remote kernels via ssh to run numerics on my cluster via my laptop. So far I've managed to get the laptop MM front end to connect via ssh to the cluster, but then it just hangs on the evaluation (after password entry). In the kernel config options I told it to...
  38. P

    Image and kernel of iterated linear transformation intersect trivially

    Homework Statement Given a linear transformation f:V -> V on a finite-dimensional vector space V, show that there is a postive integer m such that im(f^m) and ker(f^m) intersect trivially. Homework Equations The Attempt at a Solution Observe that the image and kernel of a linear...
  39. H

    Uniform Convergence of Poisson Kernel on [-π, π] minus (-a, a)

    Homework Statement show that the integral of the poisson kernel (1-r^2)/(1-2rcos(x)+r^2) converges to 0 uniformly in x as r tend to 1 from the left ,on any closed subinterval of [-pi,pi] obtained by deleting a middle open interval (-a,a) Homework Equations the integral of poisson...
  40. J

    A homomorphism is injective if and only if its kernel is trivial.

    I was a little curious on if I did the converse of this biconditonal statement correctly. Thanks in advance! =)Proposition: Suppose f:G->H is a homomorphism. Then, f is injective if and only if K={e}. Proof: Conversely, suppose K={e}, and suppose f(g)=f(g’). Now, if f(g)=f(g’)=e, then it follows...
  41. H

    Find a kernel and image basis of a linear transformation

    Homework Statement Find a kernel and image basis of the linear transformation having: \displaystyle T:{{\mathbb{R}}^{3}}\to {{\mathbb{R}}^{3}} so that \displaystyle _{B}{{\left( T \right)}_{B}}=\left( \begin{matrix} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 0 & 0 & 0 \\ \end{matrix} \right)...
  42. S

    Bases of a Linear transformation (Kernel, Image and Union ?

    Bases of a Linear transformation (Kernel, Image and Union ? http://dl.dropbox.com/u/33103477/1linear%20tran.png For the kernel/null space \begin{bmatrix} 3 & 1 & 2 & -1\\ 2 & 4 & 1 & -1 \end{bmatrix} = [0]_v Row reducing I get \begin{bmatrix} 1 & 0 & \frac{7}{9} & \frac{-2}{9}\\ 0 & 1...
  43. F

    Kernel of a Transformation that is a differential equation

    Homework Statement Calculate the kernel of https://webwork3.math.ucsb.edu/webwork2_files/tmp/equations/f7/04b646ac1797cdf54f4a373ce5ef431.png Since T is a linear transformation on a vector space of functions, your kernel will have a basis of functions. Give a basis for the kernel, you...
  44. S

    Basis of kernel and image of a linear transformation. (All worked out)

    http://dl.dropbox.com/u/33103477/linear%20transformations.png My solution(Ignore part (a), this part (b) only) http://dl.dropbox.com/u/33103477/1.jpg http://dl.dropbox.com/u/33103477/2.jpg So I have worked out the basis and for the kernel of L1 and image of L2, so I have U1 and U2...
  45. S

    Procedure for orking out the basis of the kernel of a linear transformation.

    I am working on a problem dealing with transformations of a vector and finding the basis of its kernel. Now I have worked out everything below but after reading the definitions I am a bit confused, hence just want verification if the procedure I am following is correct. My transformed matrix is...
  46. A

    MHB Can the Kernel of a Ring Homomorphism Equal 12Z or 13Z?

    Let f : Z ->C be a homomorphism of rings. Can the kernel of f be equal to 12Z or 13Z? Ok,the way I'm thinking about it is using a proof by contradiction:asuming ker f=12Z...then by the First Isomorphism Theorem for rings Z/ker f ~im f where I am f is by definition a subring of C.But since I am...
  47. S

    What is the kernel of the homomorphism defined by \theta(a+bi) = [a+3b]_{10}?

    Homework Statement 1) Show that the kernel of the homomorphism \theta: \mathbb{Z} \rightarrow \mathbb{Z}_{10} defined by \theta(a+bi) = [a+3b]_{10}, a,b \in \mathbb{Z} is <1+3i> (i.e. the ideal generated by 1+3i).The Attempt at a Solution My answer confuses me. It shows that any element...
  48. B

    Find T with Subspace S as Kernel of T

    Hi, All: I have been tutoring linear algebra, and my student does not seem to be able to understand a solution I proposed ( of course, I may be wrong, and/or explaining poorly). I'm hoping someone can suggest a better explanation and/or a different solution to this problem...
  49. V

    Need for Separate Basis for Kernel: Explained by Hello

    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...
  50. S

    Linear algebra - Image and Kernel

    Homework Statement Let V be a 3 dim vector space over F and e_1 e_2 and e_3 be those fix basis The question provide us with the linear transformation T\in L(V) such that T(e_1) = e_1 + e_2 - e_3 T(e_2) = e_2 - 3e_3 T(e_3) = -e_1 -3e_2 -2e_3 we are ask to find the matrix of T and the...
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