Kernel Definition and 200 Threads

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...

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...

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...

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?

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...

"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

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...

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...

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...

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...

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...

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...

[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...

[SOLVED] Kernel &quot;stable under&quot;: is my interpretation correct? Homework Statement A[SIZE="1"]1, A[SIZE="1"]2, A[SIZE="1"]3,..., A[SIZE="1"]r are endomorphisms. W is the kernel of A[SIZE="1"]r - lambda*I, where lambda is the eigenvalue of A[SIZE="1"]r. W is stable under A[SIZE="1"]1...

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\\...

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...

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...

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...

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...

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))...

[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...

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?

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?

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...

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 +...

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...

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)...

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...

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...

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

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...

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...

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}...

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...

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...

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...

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...

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

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...

This is probably a simple question, but just to be sure: if the kernel of linear transformation is {0}, then the set is linearly dependent since 0-vector is LD, right? So dimension is 0, right? Then what's the basis of kernel? No basis? thanks in advance.

If you have the following kernel (I think that's what it's called): A-\lambda I=\begin{pmatrix}4 & 1 \\ 4 & 1\end{pmatrix} You could write the eigenvector as: \operatorname{span}\begin{pmatrix}1 \\ -4\end{pmatrix} My question is: does it matter how you write the "span" part of it...

hi, i got a algebra question regarding kernel 1.what is the kernel of intergration operator: T(p)= p'(x)? 2. what is the kernel of differentiation operator : T(p) = integration of p(x) from -1 to 1 thanks

I've recently noticed that there is a 'Show Kernel Times' option in Windows XP's Task Manager under the 'Performance' tab. This shows up as a red meter over the green 'CPU Usage' meter. I gathered that the Kernel is a piece of software that allows the operating system to multi-task but what...

First the problem: If D_n is the Dirichlet kernel, I need to show that there exist positive constants c_1 and c_2 such that c_1 \log n \le \int\limits_{ - \pi }^\pi {\left| {D_n \left( t \right)} \right|dt} \le c_2 \log n for n=2,3,4,.... The only thing I have been able to do is this...

Show that the kernel of a field homomorphism is either the trivial homomorphism or isomorphic to the field. I've tried to see it as a factor group, but I'm stuck. Can someone help? mary

Hello i would need help to solve the integral equation with Kernel K(st)...

Hi, What's the relationship between the image and kernel of T and the image and kernel of Tn? I think we saw in class something along the lines of: Ker(T) \subseteq Ker(T^2) Im(T) \supseteq Im(T^2) My intuition is that this is also correct for any natural n, but is it true and if so...

Hi, does anyone know how to figure out the kernal and range for this linear transformation from P3 into P3 : L(p(x)) = xp'(x)? I thought ker(L)= {0} and range is P3. But the correct answer is ker (L) = P1, L(P3) = Span (x^2, x). Can someone explain to me how exactly do we fine the kernel and...

This thread is my point of view on us as Self-Award-Complex-Systems (SACS), which are part of their universe, therefore active participators in it. I think that one of the most important properties of SACS is their ability to associate between opposite things in non-destructive ways. The...

[SOLVED] Linux kernel module support How is it these days? IMO this is one of major areas where Linux used to lag behind Windows. Loading/unloading modules during runtime was not always reliable and sort of a pain... has it been improved in the last year or two?