Kernel Definition and 211 Threads

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)...

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

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 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"...

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...