Linear transformation Definition and 446 Threads

Homework Statement Let A:\mathbb R_2[x]\rightarrow \mathbb R_2[x] is a linear transformation defined as (A(p))(x)=p'(x+1) where \mathbb R_2[x] is the space of polynomials of the second order. Find all a,b,c\in\mathbb R such that the matrix \begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \\ c & 0...

Hey, I need help with part D2. My explanation is not right so I honestly do not know what I am suppose to write. My assignment is attached to this thread.

let's consider we have a linear transformation T: R^2->R^3 and alpha={ordered basis of R^2} and beta{ordered basis of R^3} and gama={v1,v2}, v1=(1,-1),v2=(2,-5). now I need to find [T]_gama(associated matrix)? When i read about it, i understood it as, first we have to find transformation of each...

1. Show that the map $\mathcal{A}$ from $\mathbb{R}^3$ to $\mathbb{R}^3$ defined by $\mathcal{A}(x,y,z) = (x+y, x-y, z)$ is a linear transformation. Find its matrix in standard basis. 2. Find the dimensions of $\text{Im}(\mathcal{A})$ and $\text{Ker}(\mathcal{A})$, and find their basis for the...

I'm confused about the notation T:R^n \implies R^m specifically about m. From my understanding if n=2 then (x1, x2). Are we transforming n=2 to another value m for example (x1, x2, x3)?

Homework Statement Consider the linear transformation T from V = P2 to W = P2 given by T(a0 + a1t + a2t2) = (−4a0 + 2a1 + 3a2) + (2a0 + 3a1 + 3a2)t + (−2a0 + 4a1 + 3a2)t^2 Let E = (e1, e2, e3) be the ordered basis in P2 given by e1(t) = 1, e2(t) = t, e3(t) = t^2 Find the coordinate matrix...

Homework Statement Find all values a\in\mathbb{R} such that vector space V=P_2(x) is the sum of eigenvectors of linear transformation L: V\rightarrow V defined as L(u)(x)=(4+x)u(0)+(x-2)u'(x)+(1+3x+ax^2)u''(x). P_2(x) is the space of polynomials of order 2. Homework Equations -Eigenvalues and...

Homework Statement Check if L(p)(x)=(1+4x)p(x)+(x-x^2)p'(x)-(x^2+x^3)p''(x) is a linear transformation on \mathbb{R_2}[x]. If L(p)(x) is a linear transformation, find it's matrix in standard basis and check if L(p)(x) is invertible. If L(p)(x) is invertible, find the function rule of it's...

Homework Statement The image of a linear transformation = columnspace of the matrix associated to the linear transformation. More specifically though, given the transformation from Rn to Rm: from subspace X to subspace Y, the image of a linear transformation is equal to the set of vectors in X...

Homework Statement Being ##f : \mathbb R^4\rightarrow\mathbb R^4## the endomorphism defined by: $$f((x,y,z,t)) = (13x + y - 2z + 3t, 10y, 9z + 6t, 6z + 4t)$$ 1) Determine the basis and dimension of ##Ker(f)## and ##Im(f)##. Complete the base chosen in ##Ker(f)## into a base of ##\mathbb R^4##...

Homework Statement Question: What is the defect of a linear transformation? 2. The attempt at a solution A defective matrix (of a linear transformation) is a matrix that doesn't have a complete basis of eigenvectors. Does this mean that linearly dependent vectors of a matrix are called defects?

Homework Statement Let A(l) = [ 1 1 1 ] [ 1 -1 2] be the matrix associated to a linear transformation l:R3 to R2 with respect to the standard basis of R3 and R2. Find the matrix associated to the given transformation with respect to hte bases B,C, where B = {(1,0,0) (0,1,0) , (0,1,1) } C =...

Homework Statement Prove that there exists only one linear transformation l: R3 to R2 such that: l(1,1,0) = (2,1) l(0,1,2) = (1,1) l(2,0,0) = (-1,-3) Find Ker(l), it's basis and dimension. Calculate l(1,2,-2) Homework EquationsThe Attempt at a Solution I still find linear transformations...

Homework Statement Given the transformation fh : R 3 → R 3 defined by fh(x, y, z) = (x−hz, x+y −hz, −hx+z), where h ∈ R is a parameter. a) Find, for all possible values of h, Ker(fh), Im(fh), their bases and dimensions. b) Is fh an isomorphism for some value of h? Homework Equations Ax=o The...

Alternate title: Is the textbook contradicting itself? imgur link: http://i.imgur.com/3sTVgwr.jpg But imgur link: http://i.imgur.com/33Ufncb.jpg So...it would appear that transposing has the property of linearity, but no matrix can achieve it...is transposing a linear transformation? The...

Mod note: Moved from Precalc section 1. Homework Statement Given l : IR3 → IR3 , l(x1, x2, x3) = (x1 + 2x2 + 3x3, 4x1 + 5x2 + 6x3, x1 + x2 + x3), find Ker(l), Im(l), their bases and dimensions. My language in explaining my steps is a little sloppy, but I'm trying to understand the process and...

Homework Statement Hi this isn't really a question but more so understanding an example that was given to me that I not know how it came to it's conclusion. This is a question pertaining linear transformation for coordinate isomorphism between basis. https://imgur.com/a/UwuACHomework Equations...

Homework Statement I've posted a few of these recently. I have one question about this problem -- hopefully my calculations are correct. f: R2 to P1, f(a,b)=b+a2x Is this a linear transformation? Homework Equations f(u+v) = f(u) + f(v) f(cu) = cf(u) where u and v are vectors in R2 and c is...

Homework Statement R2 to R1 f(x,y)=xy Determine if the transformation is linear or not Homework Equations T(V1+V2) = T(V1) + T(V2) T(cV1) = Tc(V1) The Attempt at a Solution If the function f(x,y) = xy we can define another function f(a,b)=ab Therefore, f(x,y) = f(a,b), so xy=ab, so all...

Homework Statement Homework EquationsThe Attempt at a Solution I would just like to know what is being requested when it asks me to draw sketches in order to illustrate that T is linear. Does it have something to do with altering to position of the line L itself? Any help would be very much...

Homework Statement Being T: ℝ2 → ℝ2 the linear operator which matrix in relation to basis B = {(-1, 1), (0, 1)} IS: [T]b = \begin{bmatrix} 1 & 0\\ -3 & 1 \end{bmatrix} True or False: T(x,y) = (x, 3x+y) for all x,y∈ℝ? Homework EquationsThe Attempt at a Solution 3 [/B] So first I convert (x,y)...

Homework Statement Suppose a linear transformation T: [P][/2]→[R][/3] is defined by T(1+x)= (1,3,1), T(1-x)= (-1,1,1) and T(1-[x][/2])=(-1,2,0) a) use the given values of T and linearity properties to find T(1), T(x) and T([x][/2]) b) Find the matrix representation of T (relative to standard...

Homework Statement Let T:V→V be a linear operator on a vector space V over C: (a) Give an example of an operator T:C^2→C^2 such that R(T)∩N(T)={0} but T is not a projection (b) Find a formula for a linear operator T:C^3→C^3 over C such that T is a projection with R(T)=span{(1,1,1)} and...

Homework Statement [/B] Find a 2 x 2 matrix that maps e1 to –e2 and e2 to e1+3e2Homework Equations [/B] See the above notesThe Attempt at a Solution [/B] I am making a pig's ear out of this one. I think I can get e1 to –e2 3 -1 1 -3 but as far as getting it to reconcile a matrix like...

hi guys :D im having trouble with this proof, any hints? let V be a vector space over a field F and let T1, T2: V--->V be linear transformations prove that

Let G be a non-empty open connected set in Rn, f be a differentiable function from G into R, and A be a linear transformation from Rn to R. If f '(a)=A for all a in G, find f and prove your answer. I thought of f as being the same as the linear transformation, i.e. f(x)=A(x). Is this true?

Homework Statement X ={(x1,x2,x2 −x1,3x2):x1,x2 ∈R} f(x1,x2,x2 −x1,3x2)=(x1,x1,0,3x1) 1. Find a basis for X. 2. Find dim X. 3. Find ker f and I am f 4. Find bases for ker f and I am f 5. Is f a bijection? Why? 6. Find a diagonal matrix for f.Homework EquationsThe Attempt at a Solution 1. Put...

X ={(x1,x2,x2 −x1,3x2):x1,x2 ∈R} f(x1,x2,x2 −x1,3x2)=(x1,x1,0,3x1) 1. Find a basis for X. 2. Find dim X. 3. Find ker f and I am f 4. Find bases for ker f and I am f 5. Is f a bijection? Why? 6. Find a diagonal matrix for f. My attempt: 1. (1, 1, 0, 3) and (1, 2, 1, 6) 2. Dim X = 2 3. Ker f = 0...

Homework Statement Find the domain, target space, image, rank and nullity of the linear transformation T(A)=Av, where v= (1, 2) and A is any 2×2matrix. Homework Equations The Attempt at a Solution I believe I know the domain (R2x2 vector space) and target space (R2), but I am not sure how to...

Linear transformation D:Psub2 to Psub2 defined by D( Asub0 + Asub1x + Asub2x^2) = Asub1 + 2Asub2x Find the matrix of this linear transformation with respect to the ordered bases C to C, where C= { 1-x , 1+ x, x^2 } I know that D stands for differentiating . D prime is Asub1 + 2Asub2x I...

Homework Statement Find a linear transformation w = f(z) such that it maps the disk Δ(2) onto the right half-plane {w | Re(w) > 0} satisfying f(0) = 1 and arg f'(0) = π/2 Homework Equations w = f(z) = \frac{az+b}{cz+d} z = f^{-1}(w) = \frac{dw-b}{-cw+a} The Attempt at a Solution [/B]...

Let A(a, b, c) and A'(a′,b′,c′) be two distinct points in R3. Let f from [0 , 1] to R3 be defined by f(t) = (1 -t) A + t A'. Instead of calling the component functions of f ,(f1, f2, f3) let us simply write f = (x, y, z). Express x; y; z in terms of the coordinates of A and A, and t. I thought...

Homework Statement Consider a 2x2 matrix A with A2=A. If vector w is in the image of A, what is the relationship between w and Aw? Homework Equations Linear transformation T(x)=Ax Image of a matrix is the span of its column vectors The Attempt at a Solution I know that vector w is one of the...

This is a solution that I observed from my textbook to a linear transformation problem: Isn't $T$ not linear since $\textbf{x} \ne \textbf{0}$? Property iii of the Definition of Linear Transformation states $T(\textbf{(0)} = \textbf{0}$ so something is contradictory here.

$\textbf{Problem}$ Let $\textbf{u}$ and $\textbf{v}$ be linearly independent vectors in $\mathbb{R}^3$, and let $P$ be the plane through $\textbf{u}, \textbf{v}$ and $\textbf{0}.$ The parametric equation of $P$ is $\textbf{x} = s\textbf{u} + \textbf{v}$ (with $s$, $t$ in $\mathbb{R}$). Show that...

How do you prove that rotation of a vector is a linear transformation? It's intuitive (although not completely crystal clear to me) that it is a linear transformation at the 2d level, but how do I prove it to myself (that this is a general property of rotations)? For example, rotate vector...

$\textbf{Problem}$ Given $\textbf{v} \ne \textbf{0}$ and $\textbf{p}$ in $\mathbb{R}^n$, the line through $\textbf{p}$ in the direction of $\textbf{v}$ is given by $\textbf{x} = \textbf{p} + t\textbf{v}$. Show that linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ maps this line...

Define $f: \mathbb{R} \rightarrow \mathbb{R}$ by $f(x) = mx + b$. $\textbf{a.}$ Show that $f$ is a linear transformation when $b = 0$. $\textbf{b.}$ Find a property of linear transformation that is violated when $b = 0$ $\textbf{c.}$ Why is $f$ called a linear function?

Homework Statement From Hoffman and Kunze: Is there a linear transformation T from R^3 to R^2 such that T(1,-1,1)=(1,0) and T(1,1,1)=(0,1)?Homework Equations T(c\alpha+\beta)=cT(\alpha)+T(\beta) The Attempt at a Solution I don't really understand how to prove that there is a linear...

Homework Statement t:P_3 -----> P_3 p(x) |---> p(x) + p(2) Determine whether or not this function is linear transformation or not. Homework Equations For a function to be a linear transformation then t(0) = 0 , there are other axioms that must be satisfied, but that is not the problem...

Consider the linear transformation T: R3 --> R3 /w T(v1,v2,v3)=(0, v1+v2, v2+v3) What is the preimage of w=(0,2,5) ?I tried setting up the system of equations and got v1+v2= 2 and v2+v3=5 but after that I got kinda lost in how to find the individual solutions?

Homework Statement let A be the matrix corresponding to the linear transformation from R^3 to R^3 that is rotation of 90 degrees about the x-axis Homework Equations find the matrix A The Attempt at a Solution I got stuck on rotating z component. I tried T([e1,e2,e3])=[0 -1 0]...

Homework Statement let ##T:\mathbb{R^3} \rightarrow \mathbb{R^3}## where ##T<x,y,z>=<x-2z,y+z,x+2y>## Is T one-to-one and is the range of T ##\mathbb{R^3}##? The Attempt at a Solution I took the standard matrix A ##\left[\begin{array}{cc}1&0&-2\\0&1&1\\1&2&0\end{array}\right]## det(A)=0 so...

Homework Statement Let T:R->R^2 be the linear transformation that maps the point (1,2) to (2,3) and the point (-1,2) to (2,-3). Then T maps the point (2,1) to ...Homework Equations T(xa+yb) = xT(a)+yT(b)The Attempt at a Solution Okay so I have the solution to this problem, but its understanding...

Hello guys, Let ##T: \mathbb{R^2} \to \mathbb{R^2}##. Suppose I have standard basis ##B = \{u_1, u_2\}## and another basis ##B^{\prime} = \{v_1, v_2\}## The linear transformation is described say as such ##T(v_1) = v_1 + v_2, T(v_2) = v_1## If I want to write the matrix representing ##T##...

A variant of a problem from Halmos : If AB=C and BA=D then explain why (C-D)^2 is commutative with all 2x2 matrices if A and B are 2x2 matrices. This result does not hold for any other nxn matrices where n > 2. Explain why. Edit: I tried to show that ((C-D)^2) E - E((C-D)^2) is identically...

Homework Statement Let B = {(1, -2),(2, -3)} and S be the standard basis of R2 and [-8,-4;9,4] be a linear transformation expressed in terms of the standard basis? The Attempt at a Solution 1) What is the change of basis matrix PSB ? 1,2 -2,-3 2)What is the change of...

Let L: R3 -> R3 be L(x)= \begin{pmatrix} x1+x2\\ x1-x2\\ 3x1+2x2 \end{pmatrix} find a matrix A such that L(x)=Ax for all x in R2 From what I understand I need to find the transition matrix from the elementary to L(x). However it is'nt a square matrix and it has variables instead of numbers...

I have a linear transformation, T, from P3 (polynomials of degree ≤ 3) to R4 (4-dimensional real number space). I have a second linear transformation, U, from R4 back to P3. In the first step of this four-step problem, I have shown that the composition TU from R4 to R4 is the identity linear...

Homework Statement The Attempt at a Solution I don't think I'm interpreting the question correctly. Maybe someone can point me in the right direction? There are 2 conditions: if y =/=0 then f(x,y) = x^2/y and if y=0 then f(x,y) = 0 Let u =(1,1) and v = (1,1) f(v) = f(1,1) =...