Linear transformation Definition and 446 Threads

Homework Statement f(x,y) -> |x+y| The Attempt at a Solution The answer is that the above transformation is not linear but my working shows otherwise. Here's my go: let u = (1,1) and v = (1,1) f(u) = f(1,1) = 2 f(v) = f(1,1) = 2 f(u) + f(v) = 4 f(u+v) = f(2,2) = 4...

Homework Statement Given below are three geometrically defined linear transformations from R3 to R3. You are asked to find the standard matrices of these linear transformations, and to find the images of some points or sets of points. a) T1 reflects through the yz-plane b) T2 projects...

Mod note: fixed an exponent (% --> 5) on the transformation definition. Homework Statement A is a (4x5)-matrix over R, and L_A:R^5 --> R^4 is a linear transformation defined by L_a(x)=Ax. Find the basis for the range of L_A. Homework Equations The Attempt at a Solution ##A =...

Homework Statement T4 : R3 -> R4 is defined by T4(x1, x2, x3) = (0, x1, -3 + |x1|, x1 + x2) The Attempt at a Solution I know that T4(γ1x1 + γ2x2 + γ3x3) is a linear transformation IFF γ1.T4(x1) + γ2.T4(x2) + γ3.T4(x3) T4(λ10 + λ2x1 + λ3(-3+|x1|) = λ1.T4(0) + λ2.T4(x1) +...

T2 projects orthogonally onto the xz-plane T3 rotates clockwise through an angle of 3π/4 radians about the x axis The point (-3, -4, -3) is first mapped by T2 and then T3. what are the coordinates of the resulting point? this question is on a program call Calmaeth. My answer for this...

Homework Statement Say if f is a linear transformation from R2 to R3 with f(1,0) = (1,2,3) and f(0,1) = (0,-1,2). Determine f(x,y). The Attempt at a Solution I understand the theorem on linear transformation and bases but unsure as to how I should apply it in practice. Should I be...

Homework Statement We have seen that the linear transformation ##T(x_1,x_2)=(x_1,0)## on ##\mathcal{R}^2## has the matrix ##A = \left( \begin{smallmatrix} 1&0\\ 0&0 \end{smallmatrix} \right)## with respect to the standard basis. This operator satisfies ##T^2=T##. Prove that if...

Homework Statement Homework Equations The Attempt at a Solution What is M_2 supposed to be? Is A supposed to be the matrix that produces those above linear transformations?

Homework Statement let f be the linear transformation represented by the matrix M = ( -3, 2) ( 0, -2) state what effect f has on areas, and whether f changes orientation. Find the matrix that represents the inverse of f. Homework Equations N/A The Attempt at a...

Hello, this is something basic I have hard to understand and would like to have help!:) this is a exemple from My book and I Dont understand the input! "Let T: P_2->P_2 be the linear transformation defines by T(P(x))=p(2x-1) I Dont understand how this work T(1)=1, T(x)=2x-1, T(x^2)=(2x-1)^2...

Hi everyone, :) Here's a question I encountered recently and did partway. I need your advice on how to proceed. Question: What can be said about the Jordan normal form of a linear transformation \(f:V\rightarrow V\) where \(V\) is a vector space over \(\mathbb{C}\), if we know that...

Hi everyone, :) I don't understand how to use the given linear transformation so as to calculate the exterior power of \(V\); \(\wedge^2(f)\). I hope you can help me with this. :) Problem: Find the trace of the linear transformation \(\wedge^2(f)\), if \(f\) is given by the matrix...

Let V and W be two finite-dimensional vector spaces over the field F. Let B be a basis of V, and let C be a basis of W. For any v 2 V write [v]B for the coordinate vector of v with respect to B, and similarly [w]C for w in W. Let T : V -> W be a linear map, and write [T]C B for the matrix...

I know that every linear transformation from Rn to Rm can be represented in a matrix form. What about a transformation from a 1. Infinite dimension to infinite dimension 2.finite to infinite dimension 3.infinite to finite dimension Can they represented by matrix form...? Before...

Hi everyone, :) Here's another question I encountered recently. I am writing the question and my full solution. Many thanks if you can go through it and find a mistake, or confirm whether it's correct, or can contribute with any other useful comments. :) Problem: Find the matrix of a linear...

Hi everyone, :) Here's a question I was stuck on. Hope you people can help me out. :) The definition of root vectors is given >>here<<. Now a \(n\times n\) matrix can be diagonalized if it has \(n\) distinct eigenvalues. So I don't see how the given condition (all root vectors are...

Hi everyone, :) Here's a question I got stuck. Hope you can shed some light on it. :) Of course if we write the matrix of the linear transformation we get, \[A^{t}.A=\begin{pmatrix}a_1^2 & a_{1}a_2 & \cdots & a_{1}a_{n}\\a_2 a_1 & a_2^2 &\cdots & a_{2}a_{n}\\.&.&\cdots&.\\.&.&\cdots&.\\a_n...

Hi everyone, :) This is one of those questions I encountered when trying to do a problem. I know that a eigenvector of a linear transformation should be non-zero by definition. So does that mean every linear transformation has eigenvectors? What if there's some linear transformation where no...

proof onto Prove: A linear Map T:Rn->Rm is an onto function : The only way I have thought about doing this problem is by proving the contrapositive:

Problem: A linear transformation T: Rm->Rm is invertible if and only if, for any basis {v1, ...vm} of Rm, {T(v1),...,T(vm)} is also a basis for Rm.Ideas: Since the inverse exists, we can say that some vector u in the inverse of T can be represented as linear combinations of basis vectors...

Hey This is from Numerical analysis course (Legendre polynom) - they gave us the polynomial transformation from [-1,1] to [a,b] as: x = 2/(b-a) * z - (b+a)/(b-a) what is the proof of this tranformation? where did it come from? thanks

Homework Statement If T is a linear transformation, proof that Tn is a linear transformation (with nEN). Homework Equations I know that T is a linear application if: T(u+v) = T(u) + T(v) T(au) = aT(u) The Attempt at a Solution Actually I don't know how to start using these two...

What is the difference between orthogonal transformation and linear transformation?

Resource: Linear Algebra (4th Edition) -David C. Lay I understand that there are identities associated with transformations, but what I don't understand is when the transformation is rotated about the origin through an angle β. I believe β in this case is \frac{}{}\pi/2 \left[1,0\right]...

Hello MHB, given a linear transformation F so that this is known \left\{ \begin{aligned} \phantom{1}F(1,0,0)=(1,2,3) \\ F(1,1,0)=(0,0,1)\\ F(1,1,1)=(12,3,4)\\ \end{aligned} \right. Decide F progress: F(e_1)=(1,2,3) F(e_2)=F(e_1)+F(e_2)-F(e_1)=(0,0,1)-(1,2,3)=(-1,-2,-2)...

Where T(p(x)) = (x+1)p'(x) - p(x) and p'(x) is derivative of p(x). a) Find the matrix of T with respect to the standard basis B={1,x,x^2} for P2. T(1) = (x+1) * 0 - 1 = -1 = -1 + 0x + 0x^2 T(x) = (x+1) * 1 - x = 1 = 1 + 0x + 0x^2 T(x^2) = (x+1) * 2x - x^2 = 2x + x^2 = 0 + 2x + x^2 So, the...

Hi, I'm having trouble understanding the purpose of using two basis in a linear transformation. My lecturer explained that it was a way to find a linear transformation that satisfied either dimension, but I'm having trouble understanding how that relates to the method in finding this...

Homework Statement My quantum mechanics text (in an appendix on linear algebra) states, "f the eigenvectors span the space... we are free to use them as a basis..." and then states: T|f1> = λ1f1 . . . T|fn> = λnfn My question is: is it not true that fewer than n vectors might...

Homework Statement Write down the 2 × 2 matrix that represents the following linear transformation of the plane. Also draw the image of the (first quadrant) unit square 1. T(x, y) = (2x +6y, x + 3y). Homework Equations T(x, y) = (2x +6y, x + 3y). The Attempt at a Solution So...

Homework Statement Show that the following linear transformation matrix is a contraction mapping. \begin{bmatrix} 0.5 & 0 & -1 \\ 0 & 0.5 & 1 \\ 0 & 0 & 1 \end{bmatrix} I don't know how to make that into a matrix, but it is a 3x3 matrix. The first row is [.5 0 -1] the second row is [0...

Homework Statement Define a Function T : P3 → ℝ3 by T(p) = [p(3), p'(1), 0∫1 p(x) dx ] Show that T is a linear transformation Homework Equations From the definition of a linear transformation: f(v1 + v2) = f(v1) + f(v2) and f(cv) = cf(v) The Attempt at a Solution This is how...

Homework Statement Suppose A is an m x n matrix. (a) Let v1 ,...,vn be a basis of ℝn, and Avi = wi ε ℝm, for i = 1,...,n. Prove that the vectors v1,...,vn, w1,...,wn, serve to uniquely specify A. (b) Write down a formula for A.Homework Equations Maybe B = T-1 A S The Attempt at a Solution I...

Homework Statement L: R^3 -> R^2 L(x)=(0,0)^T What is the basis, and dim of the Range? Homework Equations Rank(A)-Nullity(A)=n The Attempt at a Solution So clearly L(x)= (0,0)^T. So the basis must be the empty space and dim is zero, right? Now, going of this same logic, Say...

Let $T:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a linear transformation and $R\in \mathbb{R}^n$ be a rectangle. Prove: (1) Let $e_1,...,e_n$ be the standard basis vectors of $\mathbb{R}^n$ (i.e. the columns of the identity matrix). A permutation matrix $A$ is a matrix whose columns are...

Question Consider the linear transformation T(x1,x2,x3)= (2*x1 -2*x2- 4*x3 ,x1+2*x2+x3) (a) Find the image of (3, -2, 2) under T. (b) Does the vector (5, 3) belong to the range of T? (c) Determine the matrix of the transformation. (d) Is the transformation T onto? Justify your answer (e) Is the...

Let V be the linear space of all real polynomials p(x) of degree < n. If p ∈ V, define q = T(p) to mean that q(t) = p(t + 1) for all real t. Prove that T has only the eigenvalue 1. What are the eigenfunctions belonging to this eigenvalue? What I did was T(p)= (lamda) p = q (Lamda) p(t+1) =...

Homework Statement Let V be the linear space of all real functions Differentiable on (0,1). If f is in V define g=T(f) to mean that g(t)=tf'(t) for all t in (0,1). Prove that every real λ is an eigenvalue for T, and determine the eigenfunctions corresponding to λ.Homework Equations The Attempt...

Homework Statement |\;| is a norm on \mathbb{R}^n. Define the co-norm of the linear transformation T : \mathbb{R}^n\rightarrow\mathbb{R}^n to be m(T)=inf\left \{ |T(x)| \;\;\;\; s.t.\;|x|=1 \right \} Prove that if T is invertible with inverse S then m(T)=\frac{1}{||S||}. Homework...

$|\;|$ is a norm on $\mathbb{R}^n$. Define the co-norm of the linear transformation $T : \mathbb{R}^n\rightarrow\mathbb{R}^n$ to be $m(T)=inf\left \{ |T(x)| \;\;\;\; s.t.\;|x|=1 \right \}$ Prove that if $T$ is invertible with inverse $S$ then $m(T)=\frac{1}{||S||}$. (I think probably we need...

Homework Statement See attached images below. Homework Equations For attachment "Linear 1," I've proven that it is indeed a linear transformation. My question is what does it mean when it says to show T^2=T? What exactly is the T that I am multiplying by itself? Attachment "Linear...

Homework Statement See attached image below. Homework Equations The Attempt at a Solution I know for it to be a linear transformation it must be that: f(x)+f(y)=f(x+y) and f(tx)=tf(x) where t is a scalar. I'm not sure where to start with this proof.

Homework Statement Let V = F^n for some n ≥ 1. Show that there do not exist linear maps S, T : V → V such that ST − T S = I. The Attempt at a Solution I used induction to prove that ST^n-T^nS = nT^n-1 and that S^nT-TS^n=nS^n-1, and I know I'm supposed to use that to come up with a...

There's this theorem: A linear map T: V→W is one-to-one iff Ker(T) = 0 I'm wondering if there's an analog for showing that T is onto? If so could you provide a proof? I'm thinking it has something to do with the rank(T)...

Homework Statement Describe the possible echelon forms of the standard matrix for the linear transformation T. T: |R3 --> |R4 is one to one. The Attempt at a Solution T(x)=Ax. Right? So A must be the standard matrix. I got this: A = | £ * * | | 0 £ * | | 0 0 £ | | ? ? ? | Where £...

1. The question Let V be a vector space with the ordered basis β={v1, v2,...,vn}. Define v0=0. Then there exists a linear transformation T:V→V such that T(vj) = vj+vj-1 for j=1,2,...,n. Compute [T]β. Homework Equations [T]γβ = (aij), 1≤i≤m, 1≤j≤n (where m is dimension of γ and n is the...

I'm using a book that has a loot of errors (luckly most of them are easy to recognize, like a = instead of a ≠ or viceversa, but some are way more serious), and I'm not sure if it's a new error or a thing I don't understand. Either I didn't understood all the steps of the proof or the correct...

Here is the question: Here is a link to the question: Projection and linear transformation? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Let T be a linear transformation from R3 to R3. Suppose T transforms (1,1,0) ,(1,0,1) and (0,1,1) to (1,1,1) (0,1,3) and (3,4,0) respectively. Find the standard matrix of T and determine whether T is one to one and if T is onto Homework Equations The Attempt...

Hi, Homework Statement How may I find (or prove that there isn't) a linear transformation which satisfies T: R3->R1[x], ker T = Sp{(1,0,1), (2,-1,1)}? Homework Equations The Attempt at a Solution I am not sure how to approach this. I understand that kerT is the group of all...

Homework Statement You are given that T is a linear transformation from R^3 to P2, that T((1,1,-1)) =X, and that T((1,0,-1))=X^2+7X-1. Find T(0,-5,0) or explain why it cannot be determined form the given information. Homework Equations None The Attempt at a Solution There is only X...