Linear transformation Definition and 437 Threads

Homework Statement Let W be a complex finite dimensional vector space with a hermitian scalar product and let T: W -> W be linear and normal. Prove that U is a T-invariant subspace of W if and only if V is a T*-invariant subspace, where V is the orthogonal complement of U. The attempt at a...

If V is a vector space with an inner space <.,.>. V1 is an non empty subset of V. Vector x is contained in V is said to be orthogonal to v1 if <x,y>=0 for all y contained in V1. 1) if v is contained in V and define the mapping f(x)=<x,v>v. Show f is a linear transformation and describe its...

Homework Statement T: R3 --> R2 by T(x,y,z) = (z-x , 2y -x) v = (2, -1, -3) B = {(0,0,1),(0,1,1),(1,1,1,)} C = {(1,-1), (2,1)} What is [T]BC what is [v]B and what is T(v) Homework Equations No clue The Attempt at a Solution I found out [T]B and that's where i am stuck.

Homework Statement Determine if the following T is linear tranformation, and give the domain and range of T: T(x,y) = (x + y2, \sqrt[3]{xy} ) Homework Equations T ( u + v) = T(u) + T(v) T(ru) = rT(u) The Attempt at a Solution 1) let u = (x1, x2); T(ru ) = T(rx1, rx2)...

Homework Statement Let V be a vector space over a field F and let L(V) be the vector space of linear transformations from V to V. Suppose that T is in L(V). Do not assume that V is finite-dimensional. a) Prove that T^2 = -T iff T(x) = -x for all x in R(T). b) Suppose that T^2 = -T. Prove...

Homework Statement Let T\inL(V,V). Prove that T^{2}=0 iff T(V)\subsetn(T). Homework Equations dim T(V) + dim n(T) = dim V comes to mind. The Attempt at a Solution Honestly, I don't know where to start. I have no idea what I'm doing. My book and my professor are both utterly...

Let T: R3 -> M(2,2) be the linear transformation given by T(x,y,z) = [ z ...-z ] .....[ 0 ... x-y]Fix bases B = {(1,0,0),(0,1,0),(0,0,1)} and C = { [1 0] , [0 1] , [0 0] , [0 0] } ............[0 0]...[0 0]...[1 0]...[0 1]for R3 and M(2,2) respectivelya) Find the matrix [T]c,b of T...

First off I am NOT asking you to solve this for me. I'm just trying to understand the concept behind this problem. Let L be a linear transformation defined by L[p]=(x^2+2)p"+ (x-1)p' -4p I have not seen linear transformations in this format. Usually I see something like L(x)=x1b1+ x2b2...

Homework Statement If L( p(x) ) = [ integral (p(x)) dx , p(0) ] find representation matrix A such that L (a + bx) = A[a b]^T Homework Equations The Attempt at a Solution I don't quite understand the question. I think that: if the base from p2 is {1, x} then any...

Homework Statement Let L(x) be the Linear operator in R^2 defined by L(x) = (x1 cos a - x2 sin a, x1 sin a + x2 cos a)^T Express x1, x2 & L(x) in terms of Polar coordinates. Describe geometrically the effects of the L.T. Homework Equations Well I know that: a = tan^-1 (x2 / x1)...

Homework Statement Define T: P2-->R3 by T(p)= [p(-1)] [p(0)] [p(1)] Find the matrix for T relative to the basis {1,t,t^2} for P2 and the standard basis for R3. The Attempt at a Solution I'm not sure how to go about this. Start off by computing T(1)? But am I trying to see what...

Homework Statement Define T: R2-->R2 by T(x)=Ax Find a basis B for R2 with the property that [T]_B is diagonal. A= 0 1 -3 4 The Attempt at a Solution The eigenvalues of a diagonal matrix are its diagonal entries, so here the eigenvalues are 1, and -3. For eigenvalue=1 I get the basis...

Homework Statement Let B be an invertible n x n matrix. Prove that the linear transformation L: Mn,n \rightarrow Mn,n given by L(A) = AB, is an isomorphism. The Attempt at a Solution I know to show it is an isomorphism i need to show that L is both onto and one-to-one. By the...

Homework Statement Find a basis of the image im(LA) of the linear transformation LA: R^5 \rightarrowR^3, x\mapstoAx where A = 1 -2 2 3 -1 -3 6 -1 1 -7 2 -4 5 8 -4 and hence determine the dimension of im(LA) The Attempt at a Solution Using the equation...

if you have R2 ----> R3 and T([1,1]) = (-1,0,-3) and T([1,-1])=(5,2,-5) How can you find T([3,1]) ??

Homework Statement symmetric 2 × 2 matrices to V.Find the determinant of the linear transformation T(M)=[1,2,2,3]M+[1,2,2,3] from the space V of symmetric 2 × 2 matrices to V. Homework Equations The Attempt at a Solution hi this is my first post so if I break a rule please...

Homework Statement (a; b) is in terms of D = ( 1,1 ; 1 -1) and (c; d) is in terms of Dx = ( -1,1 ; 0,2), then we need to find a matrix such that (c;d) = (?, ?; ?, ?)* (a; b). Homework Equations y = Ax >> linear transformation The Attempt at a Solution I know the answer is [1, -3...

let A= \left( \begin{array}{Ccc} 9 & 0 \\ 2 & 6 \\ \end{array} \right) and B= \left( \begin{array}{Ccc} 5 & 1 \\ 3 & 4 \\ \end{array} \right) Find the matrix C of the linear transformation T(x)=B(A(x)). The Attempt at a Solution - Once again, I really don't know how to...

Homework Statement Let R2 => R2 be a linear transformation for which we know that: L(1,1) = (1,-2) L(-1,1) = (2,3) What is: L(-1,5) and L(a1,a2)? Homework Equations I don't know where to start. I tried writing (-1,5) as a linear combo of (1,1) and (-1,1), but that got me...

Homework Statement Check if the linear transformation f : \mathbb{R}^2 \rightarrow \mathbb{R}^2, defined with f(x,y)=(x+y,y) is isomorphism? If so, find the linear transformation f^-^1 Homework Equations V and U are vector sets. The linear copying F:V \rightarrow U which is bijection...

i need to draw 2 graphs, one arbitrary graph I make up that is not a normal distribution, and then i need to draw another in which i apply the linear transformation Y = 4X +2. I know that all the heights need to go down to 1/4 of the origional, but I don't know if it needs to shift to the right...

Homework Statement T: R2-->R2 first reflects points through -3pi/4 radian (clockwise) and then reflects points through the horizontal x1-axis. [Hint T(e1)= (-1/sqrt2, 1/sqrt2) The Attempt at a Solution I just don't understand why the points would be (-1/sqrt2, 1/sqrt2). If it's...

[SOLVED] Linear Transformation Homework Statement Determine if this is a linear transformation: L(x,y) = (x+1, y, x+y) Homework Equations This is just one, but I have no clue as to how to even begin. I've been to lecture and read the book over and over again, but i was not given...

Homework Statement I have to determine whether the following is a linear transformation T(x,y)=(x,0) Homework Equations The Attempt at a Solution again, let v=(v1, v2) and w=(w1,w2) then, T(v+w)=T(v1+w1, v2+w2)=(v1+w1, 0) and, T(v)+T(w)=(v1+w1, 0) so the first...

how do i determine whether the following is a linear transformation: T1(x,y)=(1,y) i know that it must satisfy the conditions: (a) T(v+w)=T(v)+T(w) (b) T(cv)=cT(v), where c is a real constant and v and w are real vectors in 2D. v=(v1,v2) and w=(w1,w2) but I'm still confused. Thank you

[SOLVED] Linear transformation - adding and subtracting? Homework Statement Suppose T : P2 -> P2 is a linear transformation satisfying T(3 − x + 4x^2) = 1 + x − x^2 and T(2 − 3x + 2x^2) = 7 + 3x + 2x^2. Find T(7x + 2x^2). The Attempt at a Solution First of all, it's linear. To find...

Question 1 Let T: P2 -> M22 be a linear transformation such that T(1+t)=\left[\begin{array}{cc}1&0\\0&0\end{array} \right]; T(t+t^{2})=\left[\begin{array}{cc}0&1\\1&0\end{array} \right]; T(1+t^{2})=\left[\begin{array}{cc}0&1\\0&1\end{array} \right]; Then find T(1),T(t),T(t^{2})...

Assuming that shrinking/expanding in a given direction is a linear transformation in R^3, what would be the matrix to perform it? To be more precise, given a vector e=\left(\begin{array}{c}e_1\\e_2\\e_3\end{array}\right) with a length of 1, i.e. ||e||=1 and a factor \lambda, I am...

Homework Statement Find a linear transformation T from R3 to R3 which has the effect of rotating an object clockwise by angle θ around the x-axis. Homework Equations none The Attempt at a Solution I know that I should work with matrices to show how I came up to the final matrix...

i'm studying for my midterm and I'm stumped on this section about Lienar Transformations...hope u guys can help Homework Statement question goes something like this 1) Find the standard matrix for the linear operator define by the equations (which is easy) and then determine wheter the...

[SOLVED] Linear Transformation - Linear Algebra Homework Statement Determine if T is linear. T(x,y,z) = (1,1) Homework Equations Definition of Linear Transformation: A function T: R^n --> R^m is a linear transformation if for all u and v in R^n and all scalars c, the following...

Hey i was just doping someone wouldn't mind looking over my working to see if I am on the right track! *T(x,y,z)=(-x-y-z,x+y-5z,-3x-3y+3z) is a linear transformation. S is the standard basis, S={e1,e2,e3} and B is another basis, B={v1,v2,v3} where: e1=(1,0,0) e2=(0,1,0) e3=(0,0,1) v1=(1,1,1,)...

Hi all, I have some questions about the concept of subspace of linear transformation and its dimension, when I try to prove following problems: Prove T is a finite dimensional subspace of L(V) and U is a finite dimensional subspace of V, then T(U) = {F(u) | F is in T, u is in U} is a...

Homework Statement Let V and W be vector spaces, Let T: V --> W be linear, and let {w1, w2,..., wk} be linearly independent subset of R(T). Prove that if S = {v1,v2,...vk} is chosen so that T(vi) = wi, for i = 1, 2,...,k, then S is linearly independentHomework Equations The Attempt at a...

T is a linear transformation from R^m->R^n, prove that T is continuous. I have proved that there's always a positive real number C that |T(x)|<=C|x|. How shall I proceed then? Thanks~

Hi, I'm having trouble with part two of this question. If anyone can help me out with this I would appreciate it. Thanks, Mike

1) True or False? If true, prove it. If false, prove that it is false or give a counterexample. 1a) If a linear transformation T: R^n->R^m is onto and R^n = span{X1,...,Xk}, then R^m = span{T(X1),...,T(Xk)} 1b) If T: R^n->R^m is a linear transformation and U is a subspace of R^n, then T(U)...

Homework Statement Let T: R3 --> R3 be the linear transformation that projects u onto v = (3,0,4) Find the rank and nullity of T Homework Equations So let u=(x,y,z) The Attempt at a Solution So I know that T(u) = proj. u onto v T(u) = [(3x + 4z)/ 25](3,0,4)...

Hello. I'm having some trouble with this problem. Any help would be greatly appreciated. Homework Statement Consider B= (2x+3, 3x^2 +1, -5x^2 + x-1} a) Prove that B is a basis for P_2 b) Express -x^2 - 2 as a linear combination of the elements of B c) If t: P_2 -> P_2 is a linear...

I am having some trouble with the following linear algebra problems, can someone please help me? 1) Explain what can be said about det A (determinant of A) if: A^2 + I = 0, A is n x n My attempt: A^2 = -I (det A)^2 = (-1)^n If n is be even, then det A = 1 or -1 But what happens when n...

Homework Statement Let T be a linear operator on the vector space of nxn matrices on the real field, defined by T(A) = transpose A. Show that +/- 1 are the only eigenvalues of T, and describe corresponding eigenvectors. Homework Equations The characteristic polynomial is given by...

after a series of computations, I was able to get the following matrix equation from the given of a problem: \[\left( \begin{array} {ccc} W_1 \\ W_2 \end{array} \right)\] = \[\left( \begin{array} {ccc} \frac{\sigma_{11}}{\sqrt{\sigma_{11}^2 + \sigma_{12}^2}} &...

Homework Statement Give an example of a linear transformation whose kernel is the line spanned by: -1 1 2 in lR³ Homework Equations The Attempt at a Solution Would: 1..(-1)...0 0...0...0 0..(-2)...1 be a solution?

Homework Statement The set Hom(V,W) is the collection of all linear transformations from the F-space V to the F-space W. Suppose that F,V, and W are all finite. Suppose that F=Zp for some prime p, that V is n-dimensional over F, and W is n-dimensional over F. How many elements does Hom(V,W)...

Homework Statement I have a linear map T:M(2x2) -------> M(2x2) defined by T(B) = [2 3; 4 0] * B Find a 4 × 4 matrix representation of this linear transformation with respect to the basis of M(2×2) Homework Equations T(B) = [2 3; 4 0] * B and the basis for M(2X2) is: [1 0; 0...

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}...

Ok I have to do this Linear Algebra 'Report', it is not really a Report, Report was just the best I could come up with to describe it. But anyway I have read about Vector spaces and basics and I think that I get it. Then I started reading about Linear transformation and I think it is a bit weird...

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...

Definition: Let f:V->V be a linear transformation on an inner product space V. The adjoint f* of f is a linear transformation f*:V->V satisfying <f(v),w>=<v,f*(w)> for all v,w in V. My question is would <f*(v),w>=<v,f(w)> be equivalent to the above formula in the definition? If so why...

Question: Let T:V-->W and S:W-->U be linear transformation.Show that 1) If T and S are one-to-one,then ST is one-to one 2) If ST is one-to-one,then T is one-to-one 3)Give example of two linear transformations T and S, such that ST is...