Linear transformations Definition and 200 Threads

Homework Statement Find all linear transformations ##f(z)=az+b## which map half-plane ##Im(z)>0## on ##Im(z)>0##. It is a so called self-mapping transformation. Homework Equations The Attempt at a Solution I am guessing this will have something to do with Möbius transformation...

Homework Statement Hello everyone, I have a quick question about linear transformations. In my class, we were given transformation functions and asked to decide if they are linear: The transformation defined by: T(X)= X1+X2+3 The transformation defined by: T(X)=X1+X2+(X1*X2) The...

Homework Statement Let V be the set of complex numbers regarded as a vector space over the real numbers R. Find a linear transformation T: V → V which is not complex linear (i.e. not a linear transformation if V is regarded as a vector space over the complex numbers). Homework Equations...

I've been up way too long, so pardon me if this doesn't make sense, but.. Let V and W be vector spaces. Let T and U be linear transformations from V to W. Consider the set of all x in V such that T(x) = U(x) 1.) I think that this is a subspace of V. 2.) Can I say anything about its dimension...

How do you show that a set of linear transformations from one vector space to another spans L(V,W)? This isn't a homework question, or even a question that's in the text I'm reading (Friedberg).

Homework Statement (ii) S ◦ T will be a linear transformation from P4 to R2. Write a formula for the value S(T (a4t4 + a3t3 + a2t2 + a1t + a0)) using the given formulas for T,S and use this to compute the matrix [S ◦T]B′′,B. (10p) B'' = {e1 e2} B' = {t4, t3, t2, t,1} T: P4--> M2x2 T(a4t4 +...

Hi everyone, :) Here's a question and I'll also write down the answer for which I got zero marks. :p I would really appreciate if you can find where I went wrong. Question: Let \(\phi,\,\psi\in V^{*}\) be two linear functions on a vector space \(V\) such that \(\phi(x)\,\psi(x)=0\) for all...

1. Give information Let T: P3 ---> P3 be the linear transformation described by: T(p(x))=p(x+1)+p(2-x). Find the matrix of T with respect to the standard basis b {1,x,x^2,x^3}. The Attempt at a Solution I found the transformations on the standard basis b: T(1) = 2 T(x) = 3 T(x^2) =...

Hi everyone, :) Take a look at this question. Now the problem is that I feel this question is not properly worded. If the linear transformations have rank = 1 then it is obvious that \(\mbox{Im f}=\mbox{Im g}=\{0\}\). So restating that is not needed. Don't you think so? Correct me if I am...

Here is the question: Here is a link to the question: Let {e1, e2, e3} be a basis for the vector space V and T: V -> V a linear transformation.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Say I have a linear transformation T:V##\rightarrow##W. Can I necessarily say that T(V)##\subseteq##W? I feel like T being a linear transformation would make the function behave enough to force things to not be undefined but I can't be certain..

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.

Hi, Let f be a linear transformation over some finite field, and denote f^{n} := f \circ f \circ \cdots \circ f, n times. What do we know about the linear maps f such that there exist an integer n for which f^{N} = f^n for all N \geq n? Also, how about linear maps g satisfying g = g \circ f^i...

Consider the operation of multiplying a vector in ℝ^{n} by an m \times n matrix A. This can be viewed as a linear transformation from ℝ^{n} to ℝ^{m}. Since matrices under matrix addition and multiplication by a scalar form a vector space, we can define a "vector space of linear transformations"...

I'm just having trouble understanding some of the notations given, when attempting questions such as the following: {f\inF(ℝ,ℝ): f(3)=5}. Is it just saying that, the function 'f' spans all real values?

Homework Statement Let T: R3--> R4 be a linear transformation. Assume that T(1,-2,3) = (1,2,3,4), T(2,1,-1)=(1,0,-1,0) Which of the following is T(-8,1-3)? A. (-5,-4,-3,-8) B. (-5,-4,-3,8) C. (-5,-4,3,-8). D.(-5,4,3,-8) E (-5,4,-3,8) F. None of the above.Homework Equations I really have no...

Homework Statement 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...

Why is a linear transformation T(x)=Ax one-to-one if and only if the columns of A are linearly independent? I don't get it...

This may be vague, so I apologize. I am interested in applied mathematics, so my question is about the process a scientist or engineer uses to determine what differential equation to use for a non-linear process. I am not familiar enough with describing non-linear processes to be able to...

Let T : R2 -> R2 and S : R2 -> R2 be linear transformations defined by: T(x; y) = (5x + y ; 2x + 2y) and S(x; y) = (3x + 2y ; x): (i). Find the image of the line 2x + 3y = 5 under T. (ii). Find the natural matrices of the linear transformations T o S and T^-1 Sorry, I haven't done...

My prof uses this all over his notes, and I'm still not 100% sure what he means by it: C[T]B or B[T]B From what I can gather, it has something to do with a transformation matrix, but where the B and C come into play, I have no idea.

Hey guys, I'm studying these concepts in linear algebra right now and I was wanting to confirm that my interpretation of it was correct. One to one in algebra means that for every y value, there is only 1 x value for that y value- as in- a function must pass the horizontal line test (Even...

Hi all, So this question is fairly basic, but I want to be certain I have the right idea before I do the other parts (asks about it in standard basis etc). It's a book question: Homework Statement Here are the vectors : u=[ 1 2 0] v=[2 5 0] w=[1 1 1] This forms a basis B of R3...

This thread is posted to examine the proposition that all matrices define linear transformations. But what of the matrix equation? \left[ {\begin{array}{*{20}{c}} 0 & 1 & 0 \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {blue} \\ {red} \\ {green} \\...

If we have a linear transformation T:W -> W. Then if we write T with respect to a different basis B, will the domain and range still be W? So, will we have [T]_B : W \rightarrow W ? If not, can anybody explain to me why? Thanks in advance.

Homework Statement The problem is attached. The problem statement is to "determine whether the function is a linear transformation between vector spaces." Homework Equations N/A The Attempt at a Solution T(0)=[1 0 0]^t ≠ 0, thus T is not linear. Did i do that right? It seems...

Homework Statement Today in my final i was given this exercise: Given β_1=\{v_1,v_2,v_3\} and β_2=\{u_1,u_2,u_3,u_4\}, basis of the vector spaces V and U respectively. a) Find the linear transformation T:U\rightarrow V so that T(v_i)≠T(v_j) if i≠j, T(v_1)=u_1+u_2 and T is injective b) Find...

Hi, I have to take a placement exam in linear algebra this fall so I have been studying some past exams. This is a real basic question. If we have a linear transformation T:W -> W does this imply nothing about the injectivity or surjectivity of the transformation? I assume that it does not, but...

write P for the vector space of all polynomials, a_{0}+a_{1}x+a_{2}x^{2}+...+a_{n}x^{n}, , a_{0}, a_{1},...,a_{n}\inR, n=0,1,2... 1. Find a linear transformation P->P that is onto but not one-to one 2. Find such a linear transformation, that is one-to-one but not onto I have been thinking...

Ok just for fun,could someone please give a general linear transformation of p vectors in R(n) to R(m),by expressing the transformation as a Matrix vector product of let's say n vectors in R(m).p vectors in R(n).I've already done it for fun but I'd like to see how you guys go about it..

I need some help or at least some assurance that my thinking on linear transformations and their matrix representations is correct. I assume when we specify a linear transformation eg F(x,y, z) = (3x +y, y+z, 2x-3z) for example, that this is specified by its action on the variables and is not...

http://dl.dropbox.com/u/33103477/linear%20transformations.png My attempt was to first find the transformed matrices L1 and L2. L1= ---[3 1 2 -1] -------[2 4 1 -1] L2= ---[1 -1] -------[1 -3] -------[2 -8] -------[3 -27] Now reducing L1, I have -------[1 0 7/10 -3/10]...

Hi Pf, Here is a question regard a test review that we have. I am not looking for the answer but rather a clarification about the notation. 1. What does the following mean? T1: \Re2 \rightarrow \Re2 by x \rightarrow Ax? 2. What does it mean to go \Re2 \rightarrow \Re2 Thanks.

Homework Statement Find the matrix representations [T]\alpha and [T]β of the following linear transformation T on ℝ3 with respect to the standard basis: \alpha = {e1, e2, e3} and β={e3, e2, e1} T(x,y,z)=(2x-3y+4z, 5x-y+2z, 4x+7y) Also, find the matrix representation of...

I was looking at this table here: http://en.wikipedia.org/wiki/Tensor#Examples And i didn't understand why a (1,1) tensor is a linear transformation, I was wondering if someone could explain why this is. A (1,1) tensor takes a vector and a one-form to a scalar. But a linear transformation...

Hi, Two questions: 1) Compute the matrix product corresponding to the composition of the transformations. Let U = P4(R) [polynomial degree 4], V = P3(R) , and W = P2, and let S = d/dx (derivative) and T = d/dx (derivative). Then the composition TS = d^2/dx^2 (second deriv) Attempt...

What is the most tangible way to introduce linear transformations in a linear algebra course? Most books tend to take a very abstract approach to this topic.

I was struck with the following question: Is there a linear map that's injective, but not surjective? I know full well the difference between the concepts, but I'll explain why I have this question. Given two finite spaces V and W and a transformation T: V→W represented by a matrix \textbf{A}...

Homework Statement Given the following defined transformation T(a + bt+ct^{2}) = (a+c) - (c+b)t + (a+b+c)t^{2} find the matrix with respect to the standard basis From my understanding, the standard basis for a 3 element vector would be (0,0,1)^{T} (0,1,0)^{T}...

Homework Statement Define the linear transformation T: R^{3} → R^{3} by T(v)= the projection of v onto the vector w=(1,2,1) Find the (standard matrix of T) Homework Equations T: V → W is a function from V to W (which means that for each v in V, there is a T9v) in W such that...

Homework Statement Find two linear operators T and U on R^2 such that TU = 0 but UT ≠ 0. The Attempt at a Solution Let T(x1,x2)=(0,x2) Let U(x1,x2)=(x2,0) TU(x1,x2)=T(x2,0)=(0,0) Am I right? 'Cause I can't remember if TU(x1,x2)=T[U(x1,x2)] Or TU(x1,x2)=U[T(x1,x2)]

Homework Statement Given: T is a linear transformation from V -> W and the dim(V) = n and dim(W) = m Prove: If β = {v1, ..., vm} is a basis of V, then { T(v1), ..., T(vm) } spans the image of T. NOTE: because of bad hand writing I can't tell if the bold is suppose to be an 'm' or an 'n'...

Homework Statement Show that if { v_1, ... , v_k} spans V then {T(v_1), ... , T(v_k)} spans T(v) Homework Equations The Attempt at a Solution So we know that every vector in V can be written as a linear combination of v_1,...v_k thus we only need to show that {T(v_1)...

Homework Statement Homework Equations The Attempt at a Solution In the previous problem I was asked to identify if a polynomial, such as f(x)=2x was a linear transformation. In that case I checked to see if f(ax + by) = f(ax) + f(by). I figure I would be doing something...

Homework Statement Show that 2.1.1 is equivalent to the totality of 2.1.2 and 2.1.3.Homework Equations The Attempt at a Solution aTx + bTy = aT(x) + bT(y) = T(ax) + T(by) = T(ax + by) ?

Homework Statement Let T:ℝ^{2}→ℝ be defined by T\left(\begin{array}{c} x_{1} \\x_{2}\end{array}\right) = (0 if x_{2} = 0. \frac{x^{3}_{1}}{x^{2}_{2}} otherwise.) Show that T preserves scalar multiplication, i.e T(λx) = λT(x) for all λ \in ℝ and all x \in ℝ^{2} The Attempt at a Solution...

Homework Statement Let T: P2 - P2 be the linear operator defined by T(a0 + a1x + a2x2) = a0 + a1(x - 1) + a2(x - 1)2 (a) Find the matrix for T with respect to the standard basis B = {1, x, x2}. Homework Equations [T]B[x]B = [T(x)]B The Attempt at a Solution T(1) = a0 + a1(1 -...

I have a question about mappings that go from a vector space to the dual space, the notation is quite strange. A linear functional is just a linear map f : V → F. The dual space of V is the vector space L(V,F) = (V)*, i.e. the space of linear functionals, i.e. maps from V to F. L(V,F)=...

Identify the Hermite form of the following linear transformations and the basis for its kernel (x,y,z) = (x-y+2z,2x+y-z,-3x-6y+9z) So when finding basis for kernel we have to set equal to 0, giving: x-y+2z=0 (1) 2x+y-z=0 (2) -3x-6y+9z=0...

Homework Statement My question doesn't require numerical calculation. It is more about explanation. Here it is: what does it mean to say there are unique linear transformations? My textbook says "unique linear transformations can be defined by a few values, if the given domain vectors form...