Linear transformation Definition and 437 Threads

Homework Statement find the matrices representing linear map T:R3 -> R3 with respect to the standard basis Homework Equations T reflects each point in the (x,z)-plane The Attempt at a Solution I can't figure out how does x,z plane looks like...

Homework Statement T : U -> V is a linear map defined by T(a,b) = (a-b,a+b) write down the matrix T using the standard basis of R^2 Homework Equations The Attempt at a Solution basis of V = { (1,1) , (-1,1) } standard basis of R^2 is (1,0) and (0,1) and the matrix T is...

Homework Statement Find if possible a linear transformation R^4-->R^3 so that the nullspace is [(1,2,3,4),(0,1,2,3)] and the range the solutions to x_1+x_2+x_3=0. Homework Equations - The Attempt at a Solution So I thought I should start with trying to find what kind of matrix we...

Show That There Is Only One Linear Transformation Proof Help Please! Hi, I have been trying this problem for a couple of days, I have done a proof but I don't know if it makes sense. If you want I can scan it and show it, but if someone can show me how to do it that would be more than amazing...

Homework Statement The linear operator T on R2 has the matrix \begin{bmatrix}4 & -1\\-4 & 3 \end{bmatrix} relative to the basis A = B = {(1,2), (0, 1)}. A vector u has coordinates \begin{bmatrix}1 \\1\end{bmatrix}...

Homework Statement ||T|| = {max|T(x)| : |x|<=1} show this is equivalent to ||T|| = {max|T(x)| : |x| = 1} The Attempt at a Solution {max |T(x)| : x<=1} = {max ||x|| ||T(x/||x||)|| : |x|<=1} <= {max ||T(x)|| : |x| = 1} does that look right? I need to show equality...

Homework Statement If T is a linear transformation on R^n with || T-I || < 1, prove that T is invertible. The Attempt at a Solution So a linear transformation T is invertible iff the matrix T is not singular. and I know for any matrix A, ||A|| > spectral radius(A). so, spectral...

Hello, I am working through some examples for revision purposes and am pondering over this question so would appreciate any help I could receive. I would like to prove that if T is a linear transformation on V such that T^2 = T, and I is the identity transformation on V, i)Ker(T) =...

Homework Statement Hello again. First of all thanks to anyone who has replied to my previous questions. Now, the question that troubles me is: We are given a matrix A2x2 with some random values and we are asked to say if there is a linear map which has A as its map for the standar basis...

in my book it asks me to give an example of a linear transformation T: p4 -> R^4 that's onto. I have to prove that T is onto and a linear transformation...can someone give me some advice?

Define T: F^2 --> P_1(F) by T(a, b) = a + bx (with P_1 denoting P sub 1) I usually prove problems such as this by constructing a matrix of T using bases for the vector spaces and then proving that the matrix is invertible, but is the following also a viable proof that T is an isomorphism...

Homework Statement Let v1= 1 -2 and v2= -1 1 Let T:R2R2 be the linear transformation satisfying T(v1)= 9 7 and T(v2)= 0 -8 Find the image of an arbitrary vector x y...

Homework Statement Let T: M22 -> M22 be defined by T \[ \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)\] = \[ \left( \begin{array}{cc} 2c & a+c \\ b-2c & d \\ \end{array} \right)\] Find the eigenvectors of T The Attempt at a Solution My...

Let the vectors a1,a2,a3 €R3 and b1,b2,b3 € R4 be given by a1 a2 a3 1 -2 3 2 2 1 1 1 2 b1 b2 b3 1 1 -1 2 -3 2 1 4 3 3 -2 1 The linear transformation S : R3 --> R4 is defined by...

Hi ! I am a little bit confused with notation in the following: Let A= \begin{bmatrix} 2 & 3 & 4 \\ 8 & 5 & 1 \\ \end{bmatrix} and consider A as a linear transformation mapping \mathbb{R}^3 to \mathbb{R}^2. Find the matix representation of A with respect to the bases...

Homework Statement let x=ksin(t). let k<1. let x'=x. let t'=t-kx. solve for x' as a function of t'. (this question has to do with relativity and deBroglie waves)Homework Equations given above.The Attempt at a Solution since t=t'+kx therefore x'=ksin(t'+kx). but I need x' as a function of t'...

Homework Statement We regard each polynomial p(t) an element of R(t) as defining a function p:R\rightarrow R, x \rightarrow p(x) prove that g:R[t]\rightarrow R[t], p(t) \rightarrow \int_{0}^{t}p(x)dx defines an injective linear transformation. Homework Equations The...

Homework Statement let T:V \to V be a linear transformation which satisfies T^2 = \frac{1}{2} (T + T^*) and is normal. Prove that T=T^*. Homework Equations The Attempt at a Solution I think we should start like this: Let \mathbf{A}=[T]_B be the matrix representation of T in the...

Homework Statement Define T:P_1 \rightarrow P_1 by T(a+bx)=(a-b)+ax. Check as to whether T has an inverse or not and if it has, find T^{-1} Homework Equations T^{-1}(T(v))=v The Attempt at a Solution The range of T is P1, so T is one-to-one. The inverse of T is therefore...

Homework Statement Find a linear transformation T: P_2 \rightarrow M_{22} such that T(1+x)=\left[\begin{array}{cc}1&0\\0&0\end{array}\right] T(x+x^2)=\left[\begin{array}{cc}0&1\\1&0\end{array}\right] T(1+x^2)=\left[\begin{array}{cc}0&0\\0&1\end{array}\right] The Attempt...

Homework Statement This is a slight variation of the last problem I posted. Write the standard matrix representation for T1T2 and use it to find [T1T2(1,-3,0)]E. Homework Equations T_1\left(x_1,x_2,x_3\right)=\left(x_3,-x_1,x_3\right)...

Homework Statement Write the standard matrix representation for T1 and use it to find [T1(1,-3,0)]E. Homework Equations T_1\left(x_1,x_2,x_3\right)=\left(x_3,-x_1,x_3\right) The Attempt at a Solution I just wanted to check to see if I am doing this right. Thanks in advance! A=\left(...

Homework Statement Let \mathrm{V} be a vector space. Determine all linear transformations \mathrm{T}:V\rightarrow V such that \mathrm{T}=\mathrm{T}^2. Homework Equations Hint was given and it was like this: Note that x=\mathrm{T}(x)+(x-\mathrm{T}(x)) for every x in V, and show that...

Homework Statement True or False: T(x,y)=(2x+5y,-x+2)\text{ is a linear transformation from }\mathbb{R}^2\text{ to }\mathbb{R}^2. Homework Equations None The Attempt at a Solution I thought the answer was true, but the correct answer is false. Here is my reasoning for true...

Homework Statement Let F be the vector space of all functions mapping R into R, and letT:F-F be a linear transformationsuch that T(e^2x)=x^2, T(e^3x)= sinx, and T(1)= cos5x. Find the following, if it is determined by this data. Homework Equations a. T(e^5x) b. T(3+5e^3x) c. T(3e^4x)...

Homework Statement Let T: P3-P3 be the linear transformation defined by T(p(x))= D^2(p(x))-4D(p(x)) + p(x). Find the Matrix representation of A of T, where B = (x, 1+x, x+x^2, x^3). Homework Equations The Attempt at a Solution I don't know where to start here. What is D? Is...

Homework Statement let T : R3 --> R2 be the transformation defined by T([x]) = [y+z] y x+z...

Homework Statement T: R^n \rightarrow R^m is a linear transformation. a) Calculate Dim(ran(T)) if T is one to one. b) Calculate Dim(ker(T)) if T is onto. The Attempt at a Solution a) I need to calculate the dimension of range of T if it's 1-1. So there is a property that...

Homework Statement Let D : P3--> P2 be differentiation of polyonimals. Determine the matrix of D with respect to the standard basis of P3. Homework Equations None The Attempt at a Solution I think D=[1 0 0; 0 1 0; 0 0 0]. This is from inspection though because I know that the...

Homework Statement Let L : R3[τ] → R2[τ] be a linear transformation, where the bases for the polynomial vector spaces R3[τ] and R2[τ] are (1,τ,τ2) and (1,τ) respectively. We also know the matrix representation for L is: A=[2 0 1] [0 1 3] What is the result of L(α+βτ+γτ2)...

Homework Statement Let V=<sinx,cosx> and T: V --> V be a transformation defined by T(f)=df/dx +f. Prove T is linear. The Attempt at a Solution T(f+g) = cosx-sinx+sinx+cosx T(f)+T(g) = (sinx+cosx)'+sinx+cosx = T(sinx)+T(cosx) T(αf)=αcosx +αsinx αT(f)= α(cosx+sinx)...

Homework Statement Assume that T defines a linear transformation and use the given information to find the matrix of T T: R4-->R2 such that T(1,0,0,0)=(3,-2), T(1,1,0,0)=(5,1), T(1,1,1,0)=(-1,0), and T(1,1,1,1)=(2,2)Homework Equations The Attempt at a Solution I think I need to use/find the...

Homework Statement from calculus we know that ,for any polynomial function f : R-R of degree <= n,the fuction of I(f) :R-R ,s----\intf(u) du is a polynomial function of degree <=n+1 show that the map I: Pn--Pn+1 , f--I(f) is an injective linear transformation, determine a basis of the image...

Example Let T: P_1(R) --> R^2 be the linear transformation defined by T(a + bx) = (a, a+b). The reader can verify directly that T-1: R^2 --> P_1(R) is defined by T-1(c, d) = c + (d-c)x. Observe that T-1 is also linear. I am reading my text and it kind of makes sense, but I have no clue how to...

Theorem 2.15: Let A be an m x n matrix with entries from F. Then the left-multiplication transformation L_A: F^n --> F^m. Furthermore, if B is any other m x n matrix ( with entries from F ) and B and D are the standard ordered bases for F^n and F^m, respectively, then we have the following...

How is LA a linear function? What kind of operation is action on A? I thought L denotes a linear transformation. So if we have a matrix A, how is the LA a transformation? Is it just a definition (notation wise) or is there more to it? Thanks, JL

Find a basis of Rn such that the matrix B of the given linear transformation T is diagonal. Orthogonal Projection T onto the line in R^3 spanned by (1 1 1) I'm assuming (though I tend to be wrong) that I need to find a vector that is parallel to the line and 2 that are perpendicular to...

Homework Statement Determine whether the following are linear transformations from R2 into R3: Homework Equations a) L(x)=(x1, x2, 1)^t b) L(x)=(x1, x2, x1+2x2)^t c) L(x)=(x1, 0, 0)^t d) L(x)=(x1, x2, x1^2+x2^2)^t The Attempt at a Solution To show L is a linear transformation, I need to be...

The problem: T(x + yi) = x C -> C (Complex Numbers) Show that the above is: Linear Isomorphic This is what i have for showing it's linear: T(x+yi + a + bi) = x + a + i(y + b) => T(x+a) => T(x) + T(a) T(k(x+yi)) = k(x) + k(yi) = T(kx) = kT(x) I assume that i(y+b) = 0. Is...

Homework Statement T: M22 --> M22 defined by T(A) = AB where B = [ 3 2 ] [ 2 1 ] Is the linear transformation matrix T invertible with respect to the standard bases? If so, find it. Homework Equations none The Attempt at a Solution This is going to sound stupid, but I need...

Theorem 2.12: Let A be an mxn matrix, B and C be nxp matrices, and D and E b qxm matrices. Then (d.) If V is an n-dimensional vector space with an ordered basis B, then [IV]B = In. My question: What does [IV]B mean? Is this the identity matrix with respect to the vector space V which is...

I am reading (theorem 2.14) from a textbook, and don't understand how g = Tf and (#1) line of reasoning. The theorem and proof is as follows: Theorem 2.14: Let V and W be finite-dimensional vector spaces having ordered bases B and C, respectively, and let T: V-->W be linear. Then, for...

Homework Statement Let (u,v,w) be a basis for vector space V, and let L be a linear transformation from V to vector space W. If (L(u),L(v),L(w)) is linearly dependent, then dim(Null Space(L)) > 1. Homework Equations The Attempt at a Solution I don't see why dim(Null...

Homework Statement Let T: V \rightarrow Z be a linear transformation of a vector space V onto a vector space Z. Define the mapping \bar{T}: V/N(T) \rightarrow Z by \bar{T}(v + N(T)) = T(v) for any coset v+N(T) in V/N(T). a) Prove that \bar{T} is well-defined; that is, prove that if...

Homework Statement M22 ---> R is a linear transformation. given: T[ 1 0 ] = 1 ,,[ 0 0 ] T[ 1 1 ] = 2 ,,[ 0 0 ] T[ 1 1 ] = 3 ,,[ 1 0 ] T[ 1 1 ] = 4 ,,[ 1 1 ] find T[ 1 3 ] ,,[ 4 2 ] and T[ a b ] ,,[ c d ] Homework Equations none. The Attempt at a...

Homework Statement From Calculus on Manifolds by Spivak: 1-10 If T:Rm -> Rn is a Linear Transformation show that there is a number M such that |T(h)| \leq M|h| for h\inRm Homework Equations T is a Linear Transformation => For All x,y \in Rn and scalar c 1. T(x+y)=T(x)+T(y) 2...

Homework Statement From Calculus on Manifolds by Spivak: 1-7 A Linear Transformation T:Rn -> Rn is Norm Preserving if |T(x)|=|x| and Inner Product Preserving if <Tx,Ty>=<x,y>. Prove that T is Norm Preserving iff T is Inner Product Preserving. Homework Equations T is a Linear...

Homework Statement Is T a linear transformation? T: M22 --> M22 defined as: T [ a b ] = [ 1 (a-d) ] , [ c d ] ,,, [ (b-c) 1 ] Homework Equations none. The Attempt at a Solution I need to show that it is closed under addition and scalar...

Hello everybody, I have a problem. There is a linear trasformation \xi:\mathbb{R}^2\mapsto\mathbb{R}^2 and: \xi\begin{pmatrix}3\\1\end{pmatrix}=\begin{pmatrix}2\\-4\end{pmatrix} \xi\begin{pmatrix}1\\1\end{pmatrix}=\begin{pmatrix}0\\2\end{pmatrix} How to find a matrix for this linear...

Suppose a linear transformation T: R^2 \rightarrow R^3 was defined by T(a_1,a_2) = (2a_1, a_2 + a_1, 2a_2). Now, for example, would I be allowed to evaluate T(3,8,0) by rewriting (3,8,0) as (3,8)?