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. H

    What is the Kernel, Nullity, Range, and Rank of T given a specific matrix A?

    Homework Statement let T: R^4 --->R^3, where T(v)=A(v) and matrix A is defined by A = [2 1 -1 1 1 2 0 5 4 -1 1 0 Find kernel of T, nullity of T, range of T and rank of T Homework Equations The Attempt at a Solution ok. ker(T) = Null(A)...
  2. P

    Determine if morphism, find kernel and image

    Homework Statement Determine if the following is a group morphism. Find the kernel and the image if so. f:C_{2} \times C_{3} \rightarrow S_{3} where f(h^{r},k^{s})=(1,2)^{r} \circ (123)^{s} Homework Equations The Attempt at a Solution I'm stuck on the morphism part. So I know I...
  3. T

    Kernel of a quadratic form

    Here is an interesting problem I came up with during my research. I first present a slightly simplified version. Let us the define component-wise the following bilinear symmetric form, returning a vector: a_i(u,v) = \frac{1}{2} (u^T A_i v - d_i) \;\;\; i=1 \ldots m where u,v \in V = R^n...
  4. S

    Measure theory: kernel mapping

    Let (X, \mathcal{A}), (Y, \mathcal{B}) be measurable spaces. A function K: X \times \mathcal{B} \rightarrow [0, +\infty] is called a kernel from (X, \mathcal{A}) to (Y, \mathcal{B}) if i) for each x in X, the function B \mapsto K(x,B) is a measure on (Y, \mathcal{B}), and ii) for each B in...
  5. A

    Finding the Kernel of a Complex Multiplicative Function

    Homework Statement Consider C^x, the multiplicative group of nonzero complex numbers, and let f:C^x --> C^x be defined by f(x)=x^4. Find ker f. Homework Equations C - complex numbers e^i2xpi = cos theta + isin theta element oof C R - reals Z- integers where R/Z This is the equation...
  6. K

    Linear Algebra- Kernel and images of a matrix

    Homework Statement Consider a square matrix A: a. What is the relationship between ker(A) and ker(A^2)? Are they necessarily equal? Is one of them necessarily contained in the other? More generally, What can you say about ker(A), ker(A^2), ker(A^3), ker(A^4),...? b. What can you say...
  7. J

    Can't add file containing kernel to xcode

    Hi. I'm writing a program in OpenCL but I'm very new to xcode. Basically, my program executes a kernel that exists in a separate file. I'm not sure if the code is completely correct, but the program won't build and when I do try to build it I get this error : Build GPU translate of project...
  8. B

    Basis for Image and Kernel of matrix

    Homework Statement Find an Basis for Image and Kernel of the matrix. \[ \left( \begin{array}{ccc} 2 & 1 & 3 \\ 0 & 2 & 5 \\ 1 & 1 & 1 \end{array} \right)\] Homework Equations The Attempt at a Solution To find the kernel I solve the equation Ax = 0 I put the matrix in row...
  9. Z

    Heat kernel (PDE) asymptotic expansion

    let be the PDE eigenvalue problem \partial_{t} f =Hf then if we define its Heat Kernel Z(u)= \sum_{n=0}^{\infty}e^{-uE_{n}} valid only for positive 'u' then my question is how could get an asymptotic expansion of the Heat Kernel as u approaches to 0 Z(u) \sim...
  10. B

    Integral equation with unknown kernel?

    Hi, all, I would to solve an integral equation, here is the form f(x)=\int_{x}^{R}K(x,t)g(t)dt f(x) and g(t) are known function, R is an constant, how to compute the unknown Kernel K(x,t)? Thanks a lot
  11. J

    What is the dimension of its kernel?

    Homework Statement Suppose that dim V = m and dim W = n with M>=n . If the linear map A : V -> W is onto, what is the dimension of its kernel? Homework Equations The Attempt at a Solution Onto, means that every vector in W has at least one pre-image therefore, the kernel can...
  12. J

    Kernel of the adjoint of a linear operator

    Homework Statement Let V be an inner product space and T:V->V a linear operator. Prove that if T is normal, then T and T* have the same kernel (T* is the adjoint of T). Homework Equations The Attempt at a Solution Let us assume x is in the kernel of T. Then, TT*x =T*Tx = T*0= 0...
  13. B

    Liner differentials of order n, Kernel

    Homework Statement Verify that the given function is in the kernel of L. y(x)=x-2 L = x2D2 + 2xD - 2 Homework Equations The Attempt at a Solution I took the first and 2nd derivative of y(x), and got y'(x)= -2x-3 y''(x)= 6x-4 Then plugged it into L (and a little simplifying) and got...
  14. L

    Understanding Limits of a Kernel Function

    I have a problem with my notes that I can't understand. They say: For the kernel function K_{\delta}(x)=\frac{1}{\sqrt{2 \pi \delta}} e^{-\frac{x^2}{2 \delta}} for \delta>0, we have as \delta \rightarrow 0+ , K_{\delta}(x)= \infty if x=0 and K_{\delta}(x)= 0 if x \neq 0. therefore...
  15. J

    Is Every Ideal in a Ring the Kernel of a Homomorphism?

    Kernel <--> Ideal? I know that all kernels of ring homomorphisms are ideals, but is it true that for any ideal I of a ring R, there exists a homomorphism f: R -> R' such that Ker(f)=I?
  16. S

    Linear Algebra - Basis and Kernel

    Consider a 5 x 4 matrix... We are told that the vector, 1 2 3 4 is in the kernel of A. Write v4 as a linear combination of v1,v2,v3I'm a bit confused. Since this is a kernel of A, the kernel is a subset of R^m, therefore the other columns are linear combinations and therefore redundant...
  17. L

    Dropped popcorn kernel from space will it pop?

    "If You Dropped a Corn Kernel From Space, Would it Pop During Re-Entry?" This wacky question way emailed to the magazine Popular Science insufficiently answered in Jan 2009 p.80 is it possible to figure mathematically this out without testing it
  18. D

    Finding a Matrix whose kernel is spanned by 2 vectors

    Homework Statement Find a matrix whose kernel is spanned by the two vectors u=(1,3,2) and v=(-2,0,4). Homework Equations The Attempt at a Solution Tried setting vectors as a matrix and rref'ing it, but didn't know where I was getting at, also tried using an augmented identity...
  19. J

    Kernel & Image of Linear Transformation Homework

    Homework Statement 38) Determine whether or not v1 = (-2,0,0,2) and v2 = (-2,2,2,0) are in the kernel of the linear transformation T:R^4 > R^3 given by T(x) = Ax where A = [1 2 -1 1; 1 0 1 1; 2 -4 6 2] 39) Determine whether or not w1 = (1,3,1) or w2 = (-1,-1,-2) is in...
  20. F

    Nullity, rank, image and kernel answer check

    My question is let the linear mapping T : R2->R3 be given by T(x,y)=(x-y,2y-2x,0) write down bases for its image and null-space and determine its rank and nullity. Find the matrix A that represents T with respect to the standard bases of R2 and R3 now i think i know how to do this but I'm...
  21. D

    Finding kernel of matrix transformation

    Homework Statement Find the kernel of the matrix transformation given by f(x) = Ax, where A = 1 -1 0 0 1 -2 (it's a matrix) Homework Equations Kernel is the set x in R^n for f(x) = Ax = 0The Attempt at a Solution I set up the problem like this: [ X1 X2 * A = 0 X3 ] Just...
  22. B

    Column space and kernel

    Homework Statement If col (A) is column space of A and ker(A) null space of A with ker(A) = {Ax = 0} and ker(A') = {A'y = 0} Homework Equations Consider the (3x2) matrix : A = [1,2 ; 3,4 ; 5,6] (matlab syntax) Show that col(A) = c1 * [1,0,-1]' + c2 * [0,1,2]' The Attempt...
  23. J

    What is the relationship between ker(A) and ker(A^TA)?

    B= A transpose What is the relation between ker(BA) and ker(A)? I was told that they are equal to each other, but I can't figure out why. ker(A) => Ax = 0 ker(BA) => BAx = 0 so that BA is a subset of A. This shows that ker(BA) =0 whenever ker(A) = 0, but how does this also show that...
  24. B

    Kernel and Image of Matrix AB

    [SOLVED] Kernel and Image Homework Statement Ker(A) = Im(B) AB = ? A is an m x p matrix. B is a p x n matrix. Homework Equations The Attempt at a Solution Since Ker(A) is the subset of the domain of B and Im(B) is the subset of the codomain of B, AB = I. I = identity matrix...
  25. A

    Kernel &quot;stable under&quot;: is my interpretation correct?

    [SOLVED] Kernel &quot;stable under&quot;: is my interpretation correct? Homework Statement A1, A2, A3,..., Ar are endomorphisms. W is the kernel of Ar - lambda*I, where lambda is the eigenvalue of Ar. W is stable under A1, A2, A3,..., Ar-1. Question: does "stable under" equal "closed under"...
  26. L

    Find Basis for Ker (T) with S = {1, 0, 1, 0}

    Find a basis for Ker T that contains S = \begin{pmatrix} 1\\ 0\\ 1\\ 0\\ \end{pmatrix}, \begin{pmatrix} 0\\ 1\\ 0\\ 2\\ \end{pmatrix} where T : R^4 -> R^4 is defined by T\begin{pmatrix} 1\\ b\\ c\\ d\\ \end{pmatrix} = \begin{pmatrix} a - b - c\\ a - 2b + c\\ 0\\ 0\\...
  27. S

    What is the basis of Ker(F)?

    Homework Statement Let V = M2(R) be the vector space over R of 2×2 real matrices. We consider the mapping F : V −> V defined for all matrix M belonging to V , by F(M) = AM +MA^T where A^T denotes the transpose matrix of the matrix A given below A =  1 2 −1 0  Question is...
  28. S

    Finding a Basis for the Kernel of a Differential Operator

    Homework Statement Let V = C(R,R) be the vector space of all functions f : R −> R that have continuous derivatives of all orders. We consider the mapping T : V −> V defined for all u belonging to V , by T(u(x)) = u''(x) + u'(x) − 2u(x). (Where u' is first derivative, u'' second...
  29. S

    Linear Algebra - Dimension of Kernel

    Homework Statement Suppose that U and V are finite-dimensional vector spaces and that S is in L(V, W), T is in L(U, V). Prove that dim[Ker(ST)] <= dim[Ker(S)] + dim[Ker(T)] Homework Equations (*) dim[Ker(S)] = dim(U) - dim[Im(T)] (**) dim[Ker(T)] = dim(V) - dim[Im(S)] The Attempt at a...
  30. J

    Showing that a basis for the ker(A) is in the kernel

    Homework Statement I'm new to this and I was wondering if anyone could help me out given: x+z-w=1 y-z+w=1 x+y+z=3 find the coefficient matrix A, the vector of constants B, use Gauss-jordan elimination to solve the system. Find the Rank(A), the Null(A) and a basis for the im(A) and a...
  31. N

    Kernel and images of linear operator, examples

    Homework Statement If I e.g. want to find the kernel and range of the linear opertor on P_3: L(p(x)) = x*p'(x), then we can write this as L(p'(x)) = x*(2ax+b). What, and why, is the kernel and range of this operator? The Attempt at a Solution The kernel must be the x's where L(p'(x))...
  32. N

    Kernel and image of a matrix A

    [SOLVED] Kernel and image of a matrix A Homework Statement If I have a matrix A, then the kernel of A is the solution to Ax=0? The image of A is just the vectors that span the column space? I have looked through my book and searched the WWW, but I can't find the answer to these...
  33. E

    Geometric description of kernel

    T is the projection onto the xy-coordinate plane: T(x,y,z)=(x,y,0) I have to give a geometric description of the kernel and range of T. my geometric description of the kernel: a line along the z-axis. Is this correct? whats the geometric description of the range of T?
  34. E

    Geometric description of a kernel

    let T:R^{3} \rightarrow R^{3} be a linear transformation. how can i figure out a geometric description of the kernel and range of T. What do I have to look at?
  35. C

    Proving kernel of matrix is isomorphic to 0 eigenvalue's eigenvectors

    Homework Statement I want to prove that the eigenvectors corresponding to the 0 eigenvalue of hte matrix is the same thing as the kernel of the matrix. Homework Equations A = matrix. L = lambda (eigenvalues) Ax=Lx The Attempt at a Solution Ax = 0 is the nullspace. Ax = Lx...
  36. E

    What is the kernel of a field morphism and how is it related to ideals?

    Does it make sense to talk about the kernel of a field morphism? If so, what is it? I'm getting confused because we've defined a field to be a commutative group (F,+) and a map m: F -> F s.t. (F \{0}, m) form another commutative group. For shorthand we're calling the unit element for the +...
  37. U

    What is the kernel of such a linear map

    Homework Statement This is a problem related to linear map over vector spaces of functions and finding kernels. Let V be the vector space of functions which have derivatives of all orders, and let D:V->V be the derivative. Problem1: What is the kernal of D? Problem2: Let L=D-I,where I...
  38. daniel_i_l

    What is the Dimension of the Intersection of Two Kernels in a Vector Space?

    Homework Statement Given transformations T_1, T_2:V->F where V is a vector space with the dimension n over the field F, T_1 , T_2 =/= 0. If N_1 = KerT_1 , N_2 = KerT_2 and N_1 =/= N_2 find dim(N_1 intersection N_2) Homework Equations dim(A+B) = dimA + dimB - dim(A intersection B)...
  39. R

    The Kernel of Z (mod 24) X Z (mod 81)

    Homework Statement I want to find the kernel of PHI: Z-> Z (mod 24) X Z (mod 81) I am beginning to think that the kernel of this is actually just the set containing the identity element, or the trivial subgroup of Z mod 24. I am thinking this because none of the subgroups of Z mod 24 are...
  40. J

    Find basis for the kernel of linear map

    I need help. For this problem, you have to use the Gram-Schmidt process to make it orthogonal. My trouble is finding the bais for the kernel of the linear map L: R4 -> R1 defined by L([a,b,c,d)]=a-b-2c+d I know the dimension of the kernel is 3, but how? I have tried setting it...
  41. K

    Difference between kernel f and isotrope vectors

    i have a problem in differenciating kernel f and isotrope vectors,if someone could explain me,...
  42. P

    Kernel is a principle ideal?

    Homework Statement Prove that the kernel of the homomorphism Z[x]->R sending x to 1+sqrt(2) is a principle ideal, and find a generator for this ideal. Z is the integers R is the real numbers The Attempt at a Solution I assume sending x to sqrt(2) is an example. We should first find the...
  43. T

    Express Plane V as Kernel & Image of Matrices A & B | Homework Solution

    Homework Statement Express the plane V in 3 with equation 3x1+4x2+5x3=0 as the kernel of a matrix A and as the image of a matrix B. {Note: the 1,2, and 3 after the x are subscript} Homework Equations The Attempt at a Solution Would the relevant matrix just be a [3 4 5] with an...
  44. V

    LINEAR ALGEBRA - Describe the kernel of a linear transformation GEOMETRICALLY

    Homework Statement For two nonparallel vectors \overrightarrow{v} and \overrightarrow{w} in \mathbb{R}^3, consider the linear transformation T\left(\overrightarrow{x}\right)\,=\,det\left[\overrightarrow{x}\,\,\overrightarrow{v}\,\,\overrightarrow{w}\right] from \mathbb{R}^3 to \mathbb{R}...
  45. C

    Finding range and kernel of linear transformation

    Find the range and kernel of: a) T(v1,v2) = (v2, v1) b) T(v1,v2,v3) = (v1,v2) c) T(v1,v2) = (0,0) d) T(v1,v2) = (v1, v1) Unfortunately the book I'm using (Strang, 4th edition) doesn't even mention these terms and my professor isn't helpful. My professor said: "Since range and kernel...
  46. Y

    Bounded Solution of the Heat PDE: Is u Necessarily the Heat Kernel?

    Lets say we have a solution u, to the cauchy problem of the heat PDE: u_t-laplacian(u) = 0 u(x, 0) = f(x) u is a bounded solution, meaning: u<=C*e^(a*|x|^2) Where C and a are constant. Then, does u is necesseraly the following solution: u = integral of (K(x, y, t)*f(y)) Where K...
  47. U

    How to Find Orthonormal Bases of Kernel and Row Space of Matrix A"

    A = \left(\begin{array}{cccc}-1 &6&5&9 \\ -1&0&1&3 \end{array}\right) Find orthonormal bases of the kernel, row space. To find the bases, I did reduced the array to its RREF. A = \left(\begin{array}{cccc}1 & 0&-1&-3\\ 0&1&2/3&1 \end{array}\right) Then the orthonormal bases would...
  48. S

    Kernel and image of linear transformation

    Find a basis for Ker T and a basis for I am T a) T: P_{2} -> R^2 \ T(a+bx+cx^2) = (a,b) for Ker T , both a and b must be zero, but c can be anything so the basis is x^2 for hte image we have to find the find v in P2 st T(v) = (a,b) \in P^2 the c can be anything, right? cant our basis be...
  49. N

    Finding the kernel and range of a tranformation

    If L(x) = (x1, x2, 0)^t and L(x) = (x1, x1, x1)^t What is the kernel and range?
  50. K

    Mathematica Mathematica 4.0 kernel crashes

    Greetings I have mathematica 4.0, and I've just had to install in on my laptop because my desktop HD crashed. For some reason, the kernel crashes any time I try to do a calculation, even something like 2+2 I run the same OS on my laptop as on my desktop (XP), so I have no idea what the...
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