Linear transformations Definition and 193 Threads

Homework Statement Find a basis for V. Let W be a vector space of dimension 4. Let beta = {x1, x2, x3, x4 } be an ordered basis for W. Let V = {T in L(W) | T(x1) + T(x2) = T(x4) } Homework Equations L(W) is the set of linear transformations from W to W The Attempt at a Solution...

Homework Statement Determine whether the subset W of the vector space V is a subspace of V. Let V = L(Q4) (the set of linear transformations from rational numbers with 4 coordinates to rational numbers with 4 coordinates). Let W = { T in V = L(Q4) | { (1,0,1,0) , (0,1,0,-1) } is contained in...

Homework Statement Hi all, we just started doing linear transformations in class and I still don't fully understand them. Here's one question I've been stuck on: Let P2(x,y) be the vector space of polynomials in the variables x and y of degree at most 2. Recall the monomial basis for this...

Homework Statement How do I know if this linear transformation is invertible or not? T: [ x ] ---> [ 2y ] [ y ] [ x-3y ] (I also uploaded a small .bmp file to represent this if this looks too ugly) The Attempt at a Solution Well, I thought maybe it could be...

Please, help me! Suppose n is a positive integer and T is in F^n is defined by T(z_1, z_2, ... , z_n) = (z_1+ ... +z_n, z_1+ ... +z_n, ...,z_1+ ... +z_n) Determine all eigenvalues and eigenvectors of T. Thank you in advance!

Homework Statement T: R^2-->R2 first rotates points through -3pi/4 radian clockwise and then reflects points through the horizontal x1-axis. Find the standard matrix of T. Homework Equations - The Attempt at a Solution Vector 1 is: (-1sqrt2 - 1 sqrt2) before the reflection, and...

Homework Statement Let V be the vector space of all functions f: R->R which can be differentiated arbitrarily many times. a)Let T:V->V be the linear transformation defined by T(f) = f'. Find the (real) eigenvalues and eigenvectors of T. More precisely, for each real eigenvalue describe the...

What is the common name for norm-preserving linear transformations in a normed linear space? I want to say they are the unitary transformations, but I'm just fuzzy enough not to know a good way of proving it.

Homework Statement T:{R^3 \rightarrow {R^2} given by T(v_1,v_2,v_3) = (v_3 -v_1, v_3 - v_2) If linear, specify the range of T and kernel T The attempt at a solution Okay, I went ahead and tried to find the kernel of T like here: \begin{align*}&v_3 - v_1 = 0\\ &v_3 - v_2 =...

[SOLVED] Linear transformations Homework Statement Determine whether the following maps are linear transformations. (proofs or counterexamples required) a.) L: R^2\rightarrowR^2, (x1) (x2) \mapsto (2x1 + 3x2) (0) The brackets should be two large brackets surrounding the two...

[SOLVED] Combined linear transformations Homework Statement I have a linear transformation L : R^3 -> R^3 represented by a matrix A. I also have another linear transformation S : R^3 -> R represented by a matrix B. The dimensions of the matrix A must be 3x3 and for B it is 1x3. I have to find...

Hello, Can someone help me with this problem? Thanks in advance Let T be a linear transformation such that T (v) = kv for v in R^n. Find the standard matrix for T.

Homework Statement let T: R^{3} -> R^{3} be the mapping that projects each vector x = (x(subscript 1) , x(subscript 2) , x(subscript 3) ) onto the plane x(subscript 2) = 0. Show that T is a linear transformation. Homework Equations if c is a scalar... T(cu) = cT(u) T(u + v) = T(u) +...

Homework Statement Find a basis for the image of the linear transformation T: R^4 -->R^3 given by the formula T(a,b,c,d) = (4a+b -2c - 3d, 2a + b + c - 4d, 6a - 9c + 9d) Homework Equations The Attempt at a Solution Well this question followed asking about the basis for the kernel...

If x,y of R^n (as a normed vector space) are non-zero, the angle between x and y, denoted <(x,y), is defined as arccos x.y/(|x||y|). The linear transformation T :R^n----->R^n is angle preserving if T is 1-1, and for x,y of R^n (x,y are non...

Linear transformations... Homework Statement Can't figure these things out for my life. Seriously. Here's an example. consider the basis S={v1, v2, v3} for R^3 where v1=(1,2,1), v2=(2,9,0) and v3=(3,3,4) and let T:R^3-->R^2 be the linear transformation such that: T(v1)=(1,0)...

Homework Statement Using du=.01, dv=.01 find the aroximate area under the transformation of the square bounded by the lines u=3, u=3.01, v=5, v=5.01. Homework Equations T(u,v)=<au+bv, cu+dv> where a, b, c, and d make a square matrix. The Attempt at a Solution I am not sure...

Incase anyone doesn't understand the notation, GL(2;C) is the group of linear transformations on C^2 which are invertible. Another way of looking at it is all complex 2x2 matrices with non-zero determinant. It is fairly easy to show that GL(2;C) is not simply connected (just define a...

Hi, ok I'm working with linear transformations between normed linear spaces (nls) if T :X -> Y nls's is a linear transformation, we define the norm of T, ||T||: sup{||T|| : ||x||<=1} I want to show that for X not = {0} ||T||: sup{||T|| : ||x|| = 1} frustratingly the...

1) there's given a transformation f:C^n->C^n (C is the complex field), it's known that f is linear on R (real numbers) and its rank on R equals 3 i.e, dim_R Imf=3. now is f linear on C? 2) there's a function f:C->C and its known that f is linear on R, and det_R f<0, is f linear on C? im kind...

If i am given a linear transformation D:A->A,that is followed by A=ImD(+)kerD and i am asked to prove that kerD^2=kerD and imD=imD^2. instead of trying to work it out the hard way by showing that every element of KerD is an element of kerD^2 , both directions. would it not be easier to...

Let {E1,E2,...En} be an orthogonal basis of Rn. Given k, 1<=k<=n, define Pk: Rn -> Rn by P_{k} (r_{1} E_{1} + ... + r_{n} E_{n}) = r_{k} E_{k}. Show that P_{k} = proj_{U} () where U = span {Ek} well \mbox{proj}_{U} \vec{m}= \sum_{i} \frac{ m \bullet u_{i}}{||u_{i}||^2} \vec{u} right...

"Let T be a linear transformation on a finite dimensional real vector space V. Show that T is diagonalisable if and only if there exists an inner product on V relative to which T is self-adjoint." The backward direction is easy. As for the forward direction, I don't understand how given an...

Hi the following assigment. Given P_{2} (D) be a vector space polynomials of at most degree n=2. Looking at the transformation T: P_2(D) \rightarrow D^2, where T(p) = [p(-i),p(i)]. 1) Show that this transformation is linear. I order to show this I hold my transformation up...

I have an assignment problem and I don't even know where to start... I'm taking the course through correspondence so i have no notes or prof to talk to... I've read my text and course manual over and over again but I just can't figure it out Let T: P[SIZE="1"]2->P[SIZE="1"]2 be a linear...

A question reads: Let T: V-->W be a linear transformation. a) If T is one-to-one and TR=TR1 for transformations R and R1: U -->V, show that R = R1 b) If T is onto and ST=S1T for transformations S and S1: W -->U, show that S=S1 I am sooo very lost here, and no idea where to start:(...

i think I'm just having a hard understanding linear transformations... i was asked if (5, 0) is a vector in R(T) given by the formula T(x,y)=(2x-y,-8x + 4y)...i really don't get what I'm supposed to do here.. any hints would be most appreciated.

I'm just wondering if someone can let me know if I'm on the right path here... this question asks to show that the Function T: R^3 ----> R^2 given by the formula T(X1, X2, X3) = (2X1 - X2 + X3, X2 - 4X3) is a linear transformation. soln' the definition of a L.T. is that T(u + v) = T(u)...

Hi, can someone help me get started on the following question? Q. Show that there is no line in the real plane R^2 through the origin which is invariant under the transformation whose matrix is: A\left( \theta \right) = \left[ {\begin{array}{*{20}c} {\cos \theta } & { - \sin \theta...

Okay, I will just admit that I stink at using mathematical proof in Linear. I hope someone can give me a push with this problem Prove that T : R(real)^3 -> R(real)^3 defined by T([yz,xz,zy]) is not a linear transformation. Reading my book I know that I need to prove that the...

Can anyone at least tell me how to get started on this problem I have? Problem: Determine whether the following are linear transforatmions from P2 to P3. L(p(x)) = xp(x) I understand when it's in vector form but not really picking up on the polynomial part of this.

Unique linear transformations! Problems agiain :cry: :cry: :cry: Say I have 2 vector spaces with some finite number of vectors(can assume linear independency)...how can I show that the linear transformation between the two is unique? Thanks in advance!

Hello. I am given the following: T([1,2,-3]) = [1,0,4,2] T([3,5,2]) = [-8,3,0,1] T([-2,-3,-4]) = [0,2,-1,0] And of course I know that: T(x) = Ax and I want to find the matrix A. So, from the individual equations, I construct: A[1, 2, -3] = [1, 0, 4, 2] (please forgive, these...

I've uploaded a document which I am currently working on. I would like to verify if I am doing these problems correctly. Thank you. In the first attachment (3.4b) For 1. a. 4x^3-2x b. T(P)=0 ker T={C:C \inR} Im T = {P|P is less than degree 3 or less} c. T is not one to one because P...

Linear Transformations Rn-->Rm Question I would be very grateful if someone can explain what is going on in the following problem: Determine whether the following T:Rn to Rm T(x,y)=(2x,y) Solution from solutions manual: T((x1,y1) + (x2,y2)) = (2(x1+x2), y1+y2) = (2x1,y1) + (2x2,y2)...

Prove that if T:R^{m} \rightarrow R^{n} and U:R^{n} \rightarrow R^{p} are linear transformations that are both onto, then UT:R^{n} \rightarrow R^{p} is also onto. Can anyone point me in the right direction? Is there a theorem that I can pull out of the def'n of onto that I can begin this proof?

applications ? We are studying linear transformations right now in my Lin. Alg. class. And I like to think that mathematics has some application in the real world. But what kind of appliation do matrix transfomations have? Are there any algorithms based on it? If not, it's kind of pointless in...

Im a second semester engineering student and I am a few weeks into a linear algebra class. I understand most of it, but my teacher has to work to speak english so she doesn't explain things very well. We just started linear transformations and a few things seem unclear to me. Take a shear...

Linear transformations and rotations... Hi everyone. I need some help getting started on this question. Let R: R3 ---> R3 be a rotation of pi/4 around the axix in R3. Find the matrix [R]E that defines the linear transformation R in the standard basis E={e1, e2, e3} of R3. Find R(1,2,1)...

I'm kind of stuck with the xf(0), hope this is the right place for this question? let L(f) = 2Df - xf(0) is L a linear transformation on the space of differentiable functions? thanks for your help Philip

I'd like to check my proof. It seems easy enough, but I'd like to make sure that I'm not missing anything: If V is the space of all continuous functions on [0,1] and if Tf = integral of f(x) from 0 to 1 for f in V, show that T is a linear transformation From V into R1. Like I said the...

Let g(x) belonging to Pn-1(R) be an arbiitrary polynomial of degree n-1 or less. Show that there exists a polynomial f(x) belonging to Pn(R) such that xf''(x)-f'(x)=g(x)" I interpreted this question as having to prove the linear transformation T: Pn(R) --> Pn-1(R) where f(x) |-->...

In a vector space R^3, is given a transformation A with a subscript A(x1,x2,x3)=(2*x1+x2, x1+x2+2*x3, -x2+x3). Linear transformation B has in the basis; (1,1,1), (1,0,1), (1,-1,0) a matrix T: [-1 2 3] [ 1 1 0]...