Questions tagged [linear-transformations]

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. (Def: http://en.m.wikipedia.org/wiki/Linear_map)

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Linear composition

can you help me with this quest? About composition $f$ and vector space $\mathbf{V}=\mathbb{Z^4_2}$ we know the following: $f \circ f = id_V$,$~~f $ $ \left(\begin{array}{ccc} 1\\ 0\\ 1\\ 0\\ \end{...
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1answer
35 views

Show that this operator is linear

Let $\Bbb H$ is a Hilbetr space and $T:\Bbb H\to\Bbb H$ be a operator such that $$<x,Ty>=<Tx,y>$$ $\forall x,y\in\Bbb H.$ I want to show that $T$ is linear and bounded. If I can show that $...
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892 views

How to find a matrix representing the linear transformation $xp(x)\mapsto (x-1) p(x)$?

Does there exist a matrix representation for the linear transformation $T(x P(x)) = (x-1)P(x)$, where $P(x)$ is the second degree polynomial? Here, $xP(x)$ are all third degree polynomials that ...
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Linear Algebra--searching a name for certain transformations

I am currently taking a Linear Algebra class in Spanish and having difficulty coming across the correct translation for what we are studying. I am looking at a question that asks for the rotation of ...
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1answer
113 views

Finding a formula for linear transformation

Find the formula for linear transformation $\phi : \mathbb{R}^4\rightarrow\mathbb{R}^4$ such that $\phi(\alpha_j)=\gamma_j $ for $j=1,2,3,4$ $\alpha_1 =[1,1,2,1],\alpha_2 =[1,2,1,1],\alpha_3 =[1,1,1,1]...
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2answers
75 views

Linear transformation ker and image

Let $\varphi\colon \mathbb{R}^4 \rightarrow \mathbb{R}^3$ be described by $\varphi(X)=AX$ where $A=\begin{pmatrix} 3 & 2 & 1 & 3 \\ 1 & 1 & 1 & 1 \\ 2 & 1 & 0 &...
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581 views

If $T: X\to Y$ is a bijective closed linear operator, show that $T^{-1}$ is bounded.

Let $X$ and $Y$ be normed spaces and $X$ compact. If $T: X\to Y$ is a bijective closed linear operator, show that $T^{-1}$ is bounded. I don't know where to start here. Any help would be appreciated....
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Confusion about a Linear Transformation question.

Let $\beta := [M_1, M_2, M_3, M_4]$ be the ordered basis of $R^{2×2}$ defined by: $$ M_1 := \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}, M_2 := \begin{pmatrix} 0 & 1\\ 0 & 0 \end{...
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1answer
56 views

Determining diagonalizability of a linear transformation defined by a matrix.

Suppose $A\in M_n(\Bbb C)$ satisfies $A^6-A^3+I=O$. Prove that if a linear transformation $T:M_n(\Bbb C)\rightarrow M_n(\Bbb C)$ is given by $T(B)=AB$, then $T$ is diagonalizable. How to prove it? ...
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1answer
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Prove that a function is a linear transformation.

Lets say that I have a vector space $A$ and a linear transformation defined as $f : A → A$. Now I have a function $g : A → A$ defined as $g(a) = bf(a)$ where $a\in A$ and $b \in \mathbb{R}$ is a ...
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Prove two commutative linear transformations on a vector space over an algebraically closed field can be simultaneously triangularized

Prove two commutative linear transformations on a finite-dimensional vector space $V$ over an algebraically closed field can be simultaneously triangularized. It is equivalent to show if $AB=BA$, ...
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1answer
2k views

Identity Tranformation Proof- Is this enough to prove this statement?

Let {v$_1$,...,v$_n$} be a basis for a vector space V and let T:V$\to$V be a linear transformation. Prove that if T(v$_1$)= v$_1$,...,T(v$_n$)= v$_n$, then T is the identity tranformation on V. I'm ...
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How to find dimension of vector space

In $\mathbb{R}^5$ there is given vector space $V$. Its dimension is 3. In $\mathbb{R}^{6,5}$ consider the subset $X = \{A \in \mathbb{R}^{6,5} : V \subset \ker A\}$. I have to show that $X$ is a ...
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1answer
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linear transformation and group theory

Find the image of the circle x^2+y^2=9 under the transformation : $\begin{bmatrix}2 & 4\\3 & 6\end{bmatrix}$ $\begin{bmatrix}x'\\y'\end{bmatrix}$=L$\begin{bmatrix}x\\y\end{bmatrix}$ $\begin{...
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Finding the transformation matrix given the transformation

I'm given the Transformation $P(x_1,x_2,x_3)=(x_2,x_3)$, and I'm supposed to find the transformation matrix $A$ so that $A (x_1,x_2,x_3)=(x_2,x_3)$. How do I do this? I managed to find it for a ...
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2answers
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Find a basis of the image of a linear transformation defined by: $T(a, b, c, d) = a(1 + t + t ^2 ) + b(t + t^ 2 ) + ct^2 + d$.

$T(a, b, c, d) = a(1 + t + t^2) + b(t + t^2) + ct^2 + d$ is a linear transformation. I have no idea how to go about this. Is there a way to do it without using matrices? $T: C_4 → C[t]_{≤2}$ ...
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1answer
159 views

C-vector space V linear transformation T: V → V . Show that the image + kernel is a direct sum.

A linear transformation of C-vector space (complex field) where $T: V → V$ and $T ◦ T = −2T$. $$\dim(V) = n$$ How can we prove that $\operatorname{Im}(T) + \ker(T)$ is direct? I know that i have ...
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0answers
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How do I convert between the different co-ord systems for Ray Tracing?

I am having trouble understanding how to convert (and what to convert) for my ray tracing program. At the moment I only have 3D to 2D projection transformation which is used to project the 3D world ...
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2answers
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Invertibility of Linear Transformation

Let $A$ and $B$ be two $n\times n$ matrices that have no eigenvalues in common. Show that the transformation $T$ that maps the $n\times n$ matrices, $M_n$, to the $M_n$ and is defined by the formula $$...
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1answer
2k views

Kernel and Range of a linear transformation

So the question is let T:M2x2 -> R be defined by T(A) = tr(A). Find bases for the kernel and range of the linear transformation T. Could someone explain how to solve this as I don't quite understand ...
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What exactly is Standard Coordinates?

What exactly is a standard coordinates? Sorry it seems like a very stupid question, but my professor didn't really explain it and just started to use it for solving other problems related to ...
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1answer
47 views

$r(S+T)\le r(S) + r(T)$?

If $V$ is a finite-dimensional vector space, and $S$ and $T$ are linear transformations from $V$ to $V$, how can you show that $\text{im}(S+T)$ is a subset of $\text{im}(S) + \text{im}(T)$ and also ...
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2answers
212 views

If a composition of linear transformation is invertible, then are each linear transformations invertible?

If $S$ and $T$ are linear transformations from set $V$ to $V$, which is a finite-dimensional vector space, and if the composition $ST$ is invertible, how can we show that $T$ is one-to-one, therefore, ...
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1answer
1k views

Vector Algebra Coordinate Transformation

Let us look at two coordinate systems $K$ and $K'$ with axes, respectively, $(x_1,x_2,x_3)$ and $(x_1',x_2',x_3')$ and unit vectors ($\vec{e_1},\vec{e_2},\vec{e_3}$) and ($\vec{e_1'},\vec{e_2'},\vec{...
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1answer
73 views

Transforming Vectors

Let $T$ be the linear transformation from $\mathbb{R}^3$ to $\mathbb R^3$ that reflects every vector about the $xy$-plane and then triples its length. How do I find the matrix for $T$?
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1answer
99 views

Tricky change-of-basis transformation problem

I have absolutely no idea what to do here because of the $\sin(x).$ Let $V = \text{Span}\left\{x, x^3, \sin(x) \right\}$, and consider the basis for $V$ given by $\beta = \left\{x-2x^3, x^3+\sin(x), -...
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2answers
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Determine matrix of linear transformation

Let $T:R^2\rightarrow R^2$ by $$ T \left( \begin{bmatrix} x_{1} \\ x_{2}\end{bmatrix} \right) = \begin{bmatrix} x_{2} \\ x_{1}\end{bmatrix} $$ Let A be the matrix of T. What is A. I'm having trouble ...
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1answer
343 views

Computing Hermitian Conjugate for an Operator on a Function

The operator $\hat D$ is defined by $(\hat D f)(x) = \sqrt 2 f(2x)$. Show that $\hat D$ is a linear transformation, compute its hermitian conjugate and show it is unitary. Determine all eigenfunctions ...
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1answer
126 views

linear transformation between Hilbert space

By definition, $|T|=\sup|(Tf,g)|, |f|\le1,|g|\le1$ $$||T||\ge(Tf,f)$$ But I can not find an example such that $||T||>(Tf,f)$ for any $|f|<1$. Any suggestion? Thanks in advance~
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Validity of this geometry proof

In triangle ABC, D is the midpoint of AB and E is the point of trisection of BC nearer to C. Given that $\angle$ADC = $\angle$ BAE find $\angle$BAC. Source Q5 Lets call the intersection ...
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Proof of Linear transformation

Let $T$ be a linear transformation from $\mathbb{R}^n$ to itself. For a given vector $v$ of $\mathbb{R}^n$, if $T(v)\neq 0$ but $T^2(v) = T(T(v)) = 0$, then prove that $v$ and $T(v)$ are linearly ...
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1answer
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Problem involving Derivative and Antiderivative operators in Hoffman and Kunze's Linear Algebra

I have a homework problem from Hoffman and Kunze's Linear Algebra. Let F be a subfield of $\mathbb{C}$ and let $T$, $D$ be the transformations of $F[x]$ defined by $$\begin{align} T\left(\sum_{i=0}^n{...
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1answer
862 views

Linear transformation with clockwise rotation on z axis

Let $T$ be a linear Transformation from $\mathbb{R}^3$ to itself such that $T$ is $60^{\circ}$ clockwise rotation with fixed $z$-axis (i.e, rotate the space according to the $z$-axis) where $\mathbb{R}...
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1answer
92 views

Proving linear transformation and that T is the differential operator

I have two problems that are pretty short. I understand the concepts behind, however I am not sure if my proofs are insufficient: Determine whether $T: M_{nn}\to\mathbb{R}$ defined by $T(A)=a_{11}a_{...
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1answer
378 views

Kernel, Range, and Matrix Representation of a Linear Transformation

Let L be defined on $P_3$ (the vector space of polynomials of degree less than 3) by $L(p) = q$ where $q(x) = 4p(x) − 3xp'(x) + x^2 p''(x)$. (a) Find the range of L in the form Span(. . .). (b) Find ...
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1answer
289 views

Finding a specific camera transformation matrix

I have the following situation: - two targets with known coordinates with respect to the "world". They are on a fixed xy plane on a height 0 in the z-direction. - Both targets have an angle associated ...
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3answers
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If $I-AB$ is invertible, then is $I-BA$ invertible? [duplicate]

If $A$, $B$ are square matrices and $I-AB$ is invertible how do I prove that $I-BA$ is invertible? This is exercise 8 of section 6.2 in Linear Algebra by Hoffman and Kunze. My thoughts. If $A$ and $...
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1answer
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Reducing a matrix using similarity transformations

I'm trying to reduce a matrix to an Upper Hessenberg form with similarity transformations. I figured that the Householder Method would be the way to solve this problem, but I'm having problems with ...
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Co-ordinate vector of the linear transformation of x

$T$ is the linear transformation of $V$ ($n$-dimensional) to $W$ ($m$-dimensional) and {$b_1,...b_n$} is the basis $B$ for $V.$ Given any x in $V$, the coordinate vector $[x]_B$ is in $R^n$ and the ...
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1answer
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When restriction of a diagonalizable linear operator to an eigenspace is also diagonalizable ?

Let $T, S$ be diagonalizable linear operators on $\mathbb R^n$ such that $TS = ST$. Let $E$ be an eigenspace of $T$. Is it true that the restriction of $S$ to $E$ is diagonalizable ?
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$T:\mathbb C^n \to \mathbb C^n$ is linear and $\ker(T-aI)=\ker(T-aI)^n , \forall a\in \mathbb C$ , then $T$ diagonalizable ?

If $T:\mathbb C^n \to \mathbb C^n$ is a linear transform such that $\ker(T-aI)=\ker(T-aI)^n , \forall a\in \mathbb C$ , then is $T$ diagonalizable ?
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1answer
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How to calculate the Matrix of a given Linear Transformation?

Let $V = F^3$ and $W = F^4$ and we define the following functions: $p\in {\cal L}(V,F)$ given by $p((x,y,z)) = 3x + 4y + 2z$ $q\in {\cal L}(W,F)$ given by $q((w,x,y,z)) = 2w + 5x + 7y + 11z$; $T\in ...
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Linear transformation and vectors on a certain type of 3 dimensional vector space over the rational number field

Let $V$ be a $3$ dimensional vector space over $\mathbb Q$ and $T$ be a linear transform on $V$ such that for some $\vec x , \vec y , \vec z \in V$ with $\vec x \ne \vec 0$ , $T(\vec x)=\vec y , T(\...
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1answer
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Linear Transformation: When T(v)=v

I have a homework to do in which with a pre-defined transform I have to find a vector v that after the transformation equals itself: $T(v)=v$. The transformation happens from $\mathbb{R}^3$ to $\...
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2answers
617 views

What do we mean by Derivative of linear function is a constant function.

I've the text below given in my notes: Derivative of linear function: Let $R:X\to Y$ be a linear function .Then $R':X\to L(X,Y)$ is a constant function with the constant value $R\in L(X,Y)$ i.e. $...
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If $\overline f=f-f'(a)$ then how is $\overline {f'(a)}=0$?

Below is the definition of a function being differentiable at a point, given in my notes: A function $f:A \rightarrow Y$ is said to be differentiable at $a \in A$ if there is a linear map $T \in ...
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1answer
279 views

Let $a_1, …,a_n , b_1,…b_n$ be $2n$ distinct elements of a field , then is the matrix $\Big(\dfrac1{a_i-b_j}\Big)_{ij}$ non-singular?

Let $a_1, ...,a_n , b_1,...b_n$ be $2n$ distinct elements of a field and define $$h_{ij}:=\dfrac1{a_i-b_j} , \forall i,j=1,2 ,\dots,n. $$ Is the $n \times n$ matrix $H:=(h_{ij})$ non-singular ?
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1answer
673 views

Matrix of Shift Transform on arbitrary basis

This is problem 4.4.8 of Algebra by Artin. Let $V$ be a vector space with basis $(v_1,...,v_n)$ over a field $F$, and let $a_1,...,a_{n-1}$ be elements of $F$. Define a linear operator on $V$ by the ...
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2answers
2k views

Find a matrix transformation mapping $\{(1,1,1),(0,1,0),(1,0,2)\}$ to $\{(1,1,1),(0,1,0),(1,0,1)\}$

Find a matrix transformation mapping $\{(1,1,1),(0,1,0),(1,0,2)\}$ to $\{(1,1,1),(0,1,0),(1,0,1)\}$. Is the answer $$ \begin{bmatrix}1& 0& -1\\0& 1& 1\\0& 0& 1\end{bmatrix}? ...
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1answer
268 views

Proving a transformation satisfies T(cv) = cT(v) but not T(v + w) = T(v) + T(w)

I have a doubt in the following question. Suppose $T(\vec v) = \vec v$, except that $T(0, v_2) = (0, 0)$. Show that this transformation satisfies $T(cv) = cT(v)$ but not $T(v + w) = T(v) + T(w)$. I ...