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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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Question about wording of Linear Algebra

Since I am not good at English, I am confused of what the following sentence means from T/F question. If T is linear map, then T carries linearly independent subsets of V onto linearly independent ...
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1answer
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Invariance of Matrix Entries Under Conjugation by Subsets of $GL_n$

Let $A\in\mathbb{R}^{n\times n}$ and define conjugation by $GL_n$ on $\mathbb{R}^{n\times n}$ in the usual way (e.g. for all $A\in\mathbb{R}^{n\times n}$ and $T\in GL_n$, $A\mapsto T^{-1}AT$). Are ...
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1answer
21 views

A rotation which switches two blocks of a matrix

I have a $6\times6$ matrix $A$ which I would like to transform using some unitary operator such that $B=U^\dagger AU$. I would like to swap the elements of two sub-blocks of the matrix. If the ...
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1answer
12 views

About irreducible polynomial over field & characteristic or minimal polynomial of matrix

Let $F$ be a field and $K$ be a finite extension of $F$, and let $\alpha\in K$. Consider a linear map $T:K\to K$ is defined by $T(\beta)=\alpha\beta$ for all $\beta\in K$, where $K$ viewed as a ...
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0answers
14 views

Symplectic and Euclidean structure invariance

Consider a $2n$ real symplectic space - the usual $\mathbb R^{2n}$. Suppose that the same space could be endowed too with an Euclidean structure, by which the vectors of the symplectic basis are ...
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1answer
35 views

Suppose $T: V \to W$, why matrices are used as a method of recording the values of the $Tv_j$'s in terms of a basis of $W$?

I am reading Linear Algebra Done Right Chapter 3.C It said matrices are used as an efficient method of recording the values of the $Tv_j$'s in terms of a basis of $W$. My understanding now is that ...
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2answers
41 views

Linear space of functionals from $\mathbb R^n$ to $\mathbb R$ is isomorphic to $\mathbb R^n$

Show that linear space of functionals from $\mathbb R^n$ to $\mathbb R$ i.e. $L(\mathbb R^n; \mathbb R)$ is isomorphic to $\mathbb R^n$. All I know is that I have to find bijection between these two, ...
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1answer
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Find the standard matrix, domain and codomain of the linear transformation

$T(x_1, x_2, x_3) = (x_1 -2x_2 +5x_3, 3x_1 -4x_3)$ What I have so far is: $$T\begin{bmatrix}x_1 \\ x_2 \\ x_3 \\ \end{bmatrix} = \begin{bmatrix}x_1 -2x_2 +5x_3 \\ 3x_1 -4x_3 \end{bmatrix}$$ $$T\...
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1answer
38 views

How to solve this problem raven matrices problem?

I am doing this free test in http://test.mensa.no/ That, as far as I know, the only problem I can't solve. Basically we shift the first row to the right. From first to second is easy transformation. ...
2
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1answer
29 views

How can a function including a sin operation be linearly transformable for any offset?

In the paper "Attention is all you need" the authors have chosen a function to encode the position of a word in a sequence (section 3.5). The following encoding is chosen: $ PE(pos, 2dim) = sin(pos / ...
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1answer
35 views

Does there always exist left inverses for linear transformations for finite dimensional vector spaces?

Suppose $V$ and $W$ are finite dimensional vector spaces, and that $f~:~V \to W$ is a linear map. Suppose $\{e_1, \dots, e_n\} \subset V$ and that $\{f(e_1), \dots , f(e_n)\}$ is a basis of $W$. ...
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1answer
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Is the linear map on basis of $V$ a basis of $W$?

Suppose $T \in \mathcal{L}(V,W)$, and $v_1,...v_n$ is a base of $V$, is $T(v_1),...T(v_n)$ a base of $W$? If yes, how to prove it? The reason I have this question is when I am reading the definition ...
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1answer
24 views

Extension of a linear map in a generic vector space (without Zorn's lemma)

I am studying topological vector spaces from Sevres' book "Topological vector spaces, distributions and Kernels". In one of the preparatory chapters I encountered the following excercise: Consider a ...
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1answer
26 views

Riesz representation theorem: Does the order matter?

Let $X$ be a Hilbert space. $J:X\rightarrow X',\hspace{1cm}J(x):=(\cdot,x)$ is a complex conjugated isometric isomorphism between $X$ and it's dual space $X'$. Would there be any problems as a ...
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1answer
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Linear Maps, Basis of Domain, and Matrix

I come back to study Linear Algebra Done Right from reading How to Prove It, and I am currently on 3.C Matrices. It said: We know that if $v_1,...,v_n$ is a basis of $V$ and $T:V \to W$ is linear, ...
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2answers
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Counter examples for Linear transformation from $:V\to V$

Using matrices , find examples as called for below (a) Find a linear transformation $T:V\to V$ such that $\ker T \neq 0$ but that $T$ is not surjective (b) Find a linear transformation $T:V\to V$ ...
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1answer
34 views

Is it the right way to determine the coefficients?

I am having trouble with understanding the logical completeness of a solution of exercise on my textbook (Linear Algebra Done Right). we have $Tp=(bp(1)p(2),c\sin p(0))$ &$p\in P(R)$. Prove that ...
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1answer
58 views

QR decomposition using Householder reflections: how to calculate Q?

In a recent assignment, I was asked to develop a program that could solve some specific problem using QR decomposition to find eigenvalues and eigenvectors. That assignment also specified that we ...
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1answer
31 views

Linear Transformation - One-One and Onto Property

"If T is a linear transformation on a finite dimensional vector space, then T is one-one implies T must be onto. Also T is onto implies T must be one-one." I do not understand the proof of this. ...
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1answer
44 views

Given a projection, determine if it is linear.

Let $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be a projection onto the yz-plane. Is $T$ linear? If so, find a matrix $A$ so that $T = T_A$. 1st question I have: Does it matter that $T$ is projected ...
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2answers
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Find Matrix A from Linear Transformation T

Given the following two graphics I need to matrix A so that T(x) = Ax. What I see is that 0 is a fixed point at the origin after the transformation, so I don't need to worry about it when finding ...
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0answers
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Method of approximating arbitrary unitary matrices through a universal quantum set

I have been struggling with problems asking me to construct unitary matrices out of a quantum universal set. The particular universal set I am using is $\{H, T, CNOT\}$. Where $T$ is the matrix ...
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2answers
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System of inequalities with 3 variables

I have a system of inequalities: \begin{array}{c} a-b+c>0 \\ a+b+c<4 \\ 9a-3b+c<-5 \end{array} Wolfram|Alpha says that \begin{array}{c} a < -\frac {1}{8} \end{array} But how can I ...
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1answer
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G(n) is the number of n x n matrices J with real entries that satisfy J^2 + In = 0. Show that G(n)=0 iff n is odd.

Since In is the identity matrix of order n, visualizing $J^2=-In$, how can I show that J can be a real matrix only when n is even?
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2answers
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Is $0$ the only vector in the kernel of every bounded linear functional?

Let $X$ is a normed vector space, and let $x_0\in X$ have the property that for every bounded linear functional $f:X\rightarrow K$, $f(x_0)=0$. Then does $x_0=0$? I think the answer is clearly yes, ...
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1answer
19 views

Matrix representation of a linear transformation under change of basis.

For a matrix $A\in M_n(\mathbb{F})$ consider the linear transformation $T_A:\mathbb{F}^n\rightarrow \mathbb{F}^n$ such that $x\mapsto Ax$. Suppose A is diagonalizable and $B=\{v_1,...,v_n\}$ is a ...
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1answer
32 views

Find the eigenvalue of the non-diagonalizable matrix

I have the next question: Suppose that $1$ and $2$ are eigenvalues of a linear map $\phi:\mathbb{C}^3\rightarrow\mathbb{C}^3$. Moreover, suppose that $\phi$ is not diagonalizable. Let $p=p(x)$ ...
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1answer
41 views

How does invertibility of matrix B affect the answer?

Let $V$ be vector space of $2\times2$ order matrices and let $T\in L(V,V)$ be defined as $T(A)=AB-BA$, where $B$ is $2\times2$ order invertible matrix. Then find $\dim{\ker{T}}$. I took an ...
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1answer
26 views

Show that the the linear map $I-L$ is invertible where $L:V \rightarrow V$ and $L^3 = 0$. Find the invertible matrix in terms of a polynomial.

Let $L:V \rightarrow V$ be a linear map such that $L^3 = 0$ (i.e. $L^3$ is the zero matrix). Show that $I-L$ is invertible and find $(I-L)^{-1}$ in terms of a polynomial of $L$. This question is ...
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3answers
44 views

existence of linear transformation doesn't hold

Let V and W be two vector spaces over a field F. Does there exist a linear transformation that maps some vectors in a basis of V to $0$ and two vectors in the basis to a same nonzero vector in W? I ...
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1answer
43 views

Proving the determinant for an $n\times n$ matrix equals some value

Let $n \in \mathbb{N}$. For every $1 \leq i, j, \leq n$, let $f_{ij}(x)$ be differentiable. Define the $n \times n$ matrix $A(x)$ whose $(i, j)^{\text{th}}$ entry equals $f_{ij}(x)$. Let $F(x) = \...
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3answers
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existence of linear transformation

This is a simple question in linear algebra but I just had a hard time thinking about the logic. Let V and W be two vector spaces over a field F. For simplicity, we assume they have the same finite ...
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1answer
35 views

Prove two convex sets are equal

Prove that following two sets are equal: $$ \operatorname{conv}\left\{\, xx^T \,\middle|\, x\in\Bbb R^n, \|x\|=1 \,\right\} = \left\{ A \in S_n^+ \,\middle|\, \operatorname{Tr}(A)=1 \,\right\}, $$ ...
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0answers
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Relative change with matrix operators

Is it possible to define the operator $P$ to get the following for an input matrix with all elements $a,b,c,d,e,f$ positive: $$\begin{bmatrix} a & b & c\\ d & e & f \\ \end{bmatrix} Q ...
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0answers
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Partitions of finite vector spaces into affine lines

Let $\mathbb F_q$ be the finite field with $q$ elements and $n>0$ be an integer. Let's say $M \subseteq \mathbb F_q^n$ partitions into $q^{n-1}-1$ affine lines. Hence $N := \mathbb F_q^n \setminus ...
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1answer
26 views

Derivative being linear

Very straight forward question, so I'm studying differentiation between (infinite) normed vector spaces and when considering the very basic example of $f(x)=x^2+2x$ from reals to real we have the ...
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1answer
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Prove $C^{-1} (C\mathscr{R}) \subset \mathscr{R} + ker \; C$

Prove $C^{-1} (C\mathscr{R}) \subset \mathscr{R} + ker \; C$ Given that $C:\mathscr{X} \rightarrow \mathscr{Y}$ is a linear map where $\mathscr{X} and \; \mathscr{Y}$ are finite-dimensional linear ...
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3answers
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Injectivity is equivalent to null space equals {0} [closed]

Let T : V $\rightarrow$ W. Then T is injective if and only if null T = {0}. What does the null T = {0} intuitively mean here?
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1answer
12 views

Question regarding possible implicit change of basis when performing 2D affine translation

Lets say we have a point that is a part of 3d space, but fixed to the $z=1$ plane. That is, we have, in column matrix form, assuming the canonical $\mathbb R^3$ basis: $$ v_0 = \begin{bmatrix} ...
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2answers
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Linear isometry and its trace

(We're in $\mathbb{R}^3$) What can we say about type of linear isometry $F : \mathbb{R}^3 \to \mathbb{R}^3$ if trace of $\mathrm{m} (F)$ is $-2$ or $\frac{1}{\sqrt{2}}$ or $\sqrt{2}$? Which one of ...
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1answer
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A matrix times a vectors unit vector is equal to that vector [closed]

Given the matrix A = $\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ find the vector y that satisfies A y/$|y|$ = y where $|y|$ is the norm of y
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1answer
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+50

Which functions on matrices can be represented as a matrix multiplication?

It is well known that any linear map between two finite-dimensional vector spaces, say $f: \mathbb{R}^n \to \mathbb{R}^m$, corresponds to a matrix $M \in \mathbb{R}^{n \times m}$ such that $f(x) = Mx$...
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0answers
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If $(\Phi\circ T\circ \Psi)(x) = (x_1,\cdots,x_r,0,\cdots,0)$, there exists only one choice to $(x_1,\cdots,x_r)$ such that $x\in \ker T$?

Im trying to prove the following theorem: Theorem. Let $T:\Bbb R^m \to \Bbb R^n$ be an endomorphism. So, the sentences above are equivallent: $\dim(\ker T)= r$. There exists $\Phi:\Bbb R^n\to \Bbb R^...
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1answer
29 views

Find matrices which have a specific eigenvector [closed]

Given a vector v, which is an element of a vector space V, is there a method for finding all linear operators (matrices) in which this vector is an eigenvector?
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1answer
20 views

Affine transformations that correspond to entire complex functions

I assume the following is a standard consideration and question, but I don't know how to prove it: There is a trivial one-to-one correspondence between affine transformations $f = (f_x,f_y)$ of the ...
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1answer
31 views

In the Jordan-Chevalley decomposition $M=D+N$, how obtaining $D$ and $N$ as polynomials in $M$?

The Jordan-Chevalley expresses a linear operator $M$ as $$ M = D + N, $$ where $D$ is semisimple (diagonalizable), $N$ is nilpotent and $DN=ND$. Although it is stated in many sources that $D$ and $N$ ...
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1answer
47 views

Dimensions of Ker and Im for a linear map

I'm having trouble with this problem. I know I should be using the theorem that $ \dim(\operatorname{im} T) = \dim(V) - \dim(\operatorname{Ker} T)$ to solve this question but I''m stuck on where to ...
0
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2answers
10 views

How to solve linear map with reserve?

There's linear map $L: \mathbb{R}^2 \to \mathbb{R}^2$ with reserves: $L([1,2])=[1,1], \ \ \ \ L([2,2])=[2,1]$ a) calculate $L([1,0])$ and $L([0,1])$ b) determine $L([x,y])$ I don't know how to ...
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0answers
11 views

Find best chain of transformations for fitting point sets received at different rates.

Consider a set of points $\mathbf p_{k-1} $ that are paired with another set of points $\mathbf q_{k} $ at the instant $\{k\}$. In order to find the transformation $\mathbf T_{k-1}^{k}$ that best ...
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0answers
26 views

How to calculate angle between vectors between different bases?

I have one vector $ \vec{[v]_e} = (2,1,1)^T $ in natural base and transformation $ T(x,y,z) = (x-y,y-z,z-x) $ in natural base I also have two bases $$ B = \begin{pmatrix} 1 & 1 & 0 \\...