Kernel Definition and 200 Threads

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...

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...

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...

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...

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

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...

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...

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...

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?

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...

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...

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...

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...

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!

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...

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...

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) ##...

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...

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...

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...

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...

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...

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?

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...

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...

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 +...

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

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

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...

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...

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...

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...

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...

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)...

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...

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...

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...

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...

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...

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...

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...

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...

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...

Anyone has any idea about how to emulate a SUN3 or SUN4 kernel in cygwin under windows 7? I want to run these two softwares SUPREM IV GS and SEDAN III. The are free to download and use from http://www-tcad.stanford.edu/ It seems from the makefile that it uses sun3 / sun4 architecture and was...

Homework Statement Can you look at Poisson's formula for a half plane as a limit case of Poisson's formula for a disk? http://en.wikipedia.org/wiki/Poisson_kernel I can find lots of information about the Poisson kernel for a disk, but not for the half plane. I do know on can mat the unit...

Homework Statement I am having lots of trouble understanding how to get the kernel of linear transformations. I get that you basically set it equal to zero and solve. T: P3 → P2 given by T(p(x)) = p΄΄(x) + p΄(x) + p(0) Find ker(T) The Attempt at a Solution So P3 = ax^3 + bx^2 +...

If we have a matrix M with a kernel, in many cases there exists a projection operator P onto the kernel of M satisfying [P,M]=0. It seems to me that this projector does not in general need to be an orthogonal projector, but it is probably unique if it exists. My question: is there a standard...

Homework Statement For the linear transformation T: R4 --> R3 defined by TA: v -->Av find a basis for the Kernel of TA and for the Image of of TA where A is 2 4 6 2 1 3 -4 1 4 10 -2 4Homework Equations Let v = a1 b1 c1 a2 b2 c2 a3 b3 c3 a4 b4 c4 The...

Homework Statement Matrix A = 0 1 0 0 0 1 12 8 -1 Let E1 = a(A)(A+2I)2 Let E2 = b(A)(A-3I) For each of these, calculate the image and the kernel Homework Equations I found a(A) to be 1/25 and b(A) to be 1/25*(A-7I) Also, if I am not mistaken, I think KernelE1 =...

Homework Statement If T:V\rightarrow V is linear, then Ker(T^2)=Ker(T) implies Im(T^2)=Im(T). Homework Equations Let T:V\rightarrow V be a linear operator such that \forall x\in V, T^2(x)=0\Rightarrow T(x)=0 (Ker(T^2)=Ker(T)). Prove that \forall x\in V, \exists u\in V\ni...