Skip to main content

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)

Filter by
Sorted by
Tagged with
0 votes
0 answers
26 views

Is there a counter example to disprove the following regarding vector addition in binary field?

Let $\{\mathbf{a}_1 , \mathbf{a}_2 , \mathbf{a}_3 , ...., \mathbf{a}_{30}\}\subset \mathbb{F}_2^{15}$ denote the set of binary vectors. Define the set of integers $\{p_k\}_{k=1}^{14}$ as$3 \leq p_1 &...
Dark Forest's user avatar
0 votes
0 answers
20 views

Mapping polynomial into quadratic space

Given a parametric polynomial function $y = f(x,a)$, is there a way to find a quadratic equation that has the roots of the $N/2$'th and $N/2+1$'th (sorted) root of the original polynomial, regardless ...
frgoe's user avatar
  • 31
0 votes
0 answers
11 views

cooperative games, shapley and additivity

I am studying Shapley values and am interested in understanding cases where additivity does not hold in cooperative games. Specifically, I am looking for a practical example of two cooperative games ...
volperossa's user avatar
1 vote
1 answer
23 views

Angle of rotation which is the composition of two Householder transformations in $\Bbb R^4$

Let $v_1,v_2\in \Bbb R^4$ be unit vectors with $\langle v_1,v_2\rangle =\cos \theta$. Let $H_i=I-2v_iv_i^t$ be the Householder transformation for $i=1,2$. I am asked to compute the angle of the ...
blancket's user avatar
  • 1,802
0 votes
0 answers
12 views

mirror image of a point reflected a plane [closed]

we have a point p = (0, 1)T ∈ R2 and the symmetry plane given by H = span( 1 1)T Find the coordinates of p′, the position when p is reflected at H.
Roselina Moven's user avatar
0 votes
2 answers
76 views

Prove that a square rank $1$ matrix can have at most one eigenvalue different from $0$

I'm sure this can be proven in other ways, but I'm curious if the gap in the line of reasoning presented below can be filled so that the proof is valid. Proof by contradiction. Assume the opposite: ...
powerline's user avatar
  • 537
0 votes
0 answers
31 views

Linear mapping with rational coefficient representing integer vectors

The following question arose from trying to understand systems of linear equations over integers. Given $A\in M_{m\times n}(\mathbb{Q})$ and $b\in\mathbb{Q}^m$, how to determine if there exists $x\in\...
QMath's user avatar
  • 427
1 vote
1 answer
64 views

Tensor Notation with Basis in Differential Geometry

Let's say we have two smooth riemannian manifolds $\mathfrak{B}$ and $\mathfrak{S}$ and with coordinates $X^A$ on $\mathfrak{B}$ and $x^a$ on $\mathfrak{S}$, with $A,a \in \{1,2,3\}$ Let's now assume ...
Noiv's user avatar
  • 51
0 votes
0 answers
20 views

From cartesian to (discrete) skewed coordinate system

Some context. Let's say that I have a device made of "strips" for measuring positions. By strip device I mean that I have an area $H\times W$ (see figure a) which is made of elements (the ...
lll's user avatar
  • 23
0 votes
0 answers
13 views

How to constrain a general mapping (from a multiset to a multiset) to a multirelation?

A multiset is a function, that assign a multiplicity (non-negative integer) to all elements of an underlying (universe) set. Let $A$ be a set, then $m: A \to \mathbb N_0$ is a multiset. $\mu A$ ...
Minop's user avatar
  • 101
1 vote
0 answers
11 views

Is sequence $x_n = A^nBC^n x$ either bounded or distributed linearly?

Given 3 invertible matrices $A,B,C \in \mathbb R^{k\times k}$ and a starting vector $x \in \mathbb R^k$, $x\not= 0$, define a sequence $(x_n)$ by $x_n = A^nBC^n\cdot x$. Is it always the case that $\|...
Cecilia's user avatar
  • 648
0 votes
0 answers
23 views

Graphical Intuition of a Linear Transformation in terms of Row Vectors

The graphical intuition of a linear transformation (matrix) $A \in \mathbb{R}^{m \times n}$ applied on a vector $\textbf{v}$ in terms of the column vectors $\textbf{c}_i$ of $A$ is quite clear to me: ...
olives's user avatar
  • 1
-1 votes
1 answer
28 views

Stability of Subspaces under a Linear Map in Direct Sum Decomposition

Consider the vector spaces $D_1$, $D_2$, $D$ and $X$ such that $D\subset X$ and $D=D_1\oplus D_2$. Furthermore, suppose that $L:X\longrightarrow D$ is a linear map such that $D_1$ is stable under $L$...
amine's user avatar
  • 87
1 vote
0 answers
61 views

Loomis and Sternberg Chapter 2, problem 2.19: defining the degree of a polynomial on a vector space (over R)

Exercise 2.19 of chapter 2 of L&S is: A polynomial on a vector space V is a real-valued function on V which can be represented as a finite sum of finite products of linear functionals. Define the ...
user1349439's user avatar
1 vote
3 answers
48 views

endomorphism and tensors

Can someone explain to me how an endomorphism can be related to a tensor $T_{1,1}$? I don't understand how a tensor, which a priori takes a linear functional and a vector and produces a scalar, can be ...
JL14's user avatar
  • 45
0 votes
1 answer
28 views

Transformation of contraints given non-invertible transformation of variables

Given the transformation $Ax = y$ and constraints $Cx \le d$, how to obtain the resulting constraints on $y$ when $A$ is rectangular with unknown rank? I'm thinking along the lines of using the pseudo-...
Muhammad Usama Zubair's user avatar
2 votes
0 answers
45 views

Does the trace of an operator commute with time derivatives of an operator?

I want to find the rate of entropy production in a quantum system using von Neumann entropy $$S = -tr{(\rho \ln{\rho})}$$ by taking it's time derivative. Can I take the derivative inside the trace or ...
wednesdaypotter's user avatar
2 votes
2 answers
179 views

Exercise 5.B.7(b) in "Linear Algebra Done Right" 4th edition by Sheldon Axler

The following exercise is part (b) of exercise number 7 in Sheldon Axler's Linear Algebra Done Right, 4th edition: Suppose $V$ is finite-dimensional and $S, T \in \mathcal{L}(V)$. Prove that if at ...
Paul Ash's user avatar
  • 1,454
1 vote
1 answer
71 views

Trouble finding norm of $T: H_1 \to H_2$, which is defined by $T(x)=\sum_{i=1}^n\lambda_i \langle x, a_i \rangle b_i \quad \text{for each } x\in H_1$.

Let $H_1$ and $H_2$ be complex Hilbert spaces. Let $\lambda_1, \lambda_2, \ldots, \lambda_n$ be complex numbers, and let $\{a_1, a_2, \ldots, a_n\} \subset H_1$ and $\{b_1, b_2, \ldots, b_n\} \subset ...
user avatar
0 votes
0 answers
65 views

For each $x\in X$, there exists $\lim_{n \to \infty}T_n x$ and introduce $T:X\to X$ as $Tx=\lim_{n \to \infty}T_n x\quad \text{for each } x\in X$.

Let $X$ be a Banach space and let $T_n \in \mathcal{B}(X)$ for each $n \in \mathbb{N}$. Assume that for each $x \in X$, there exists $\lim_{n \to \infty} T_n x$ and introduce the mapping $T : X \to X$ ...
user avatar
1 vote
2 answers
73 views

Exponentiation of a linear operator

I am trying to go through "Introduction to Functional Analysis" from MITOCW (MATH 18.102) by myself and I am confused by a question in the second problem set. Let $B$ be a Banach space. Let $...
Treely's user avatar
  • 13
0 votes
1 answer
23 views

Galileo transformation group

I am reading the book "Mechanics" by Florian Scheck, more specifically on Galileo's transformations. The author states, in paragraph 1.13 if anyone has the text, that the more general ...
Nameless's user avatar
0 votes
1 answer
93 views

Can $\text{rank} (T) + \text{nullity} (T) = \dim V$ be proven with this simple argument?

I am helping one of my friends with linear algebra and gave him this theorem to prove as an exercise: Theorem . Let $V$ and $W$ be vector spaces over the field $F$ and let $T$ be a linear ...
Mathematics enjoyer's user avatar
1 vote
0 answers
34 views

Existence of an orthogonal matrix $Q\in O(n)$ such that $f(A) = QAQ^T$ where $A\in Sym(n)$

Let $Sym(n)$ be the set of all $n\times n$ real symmetric matrices. Let $T: \mathbb{R}^{n\times n} \rightarrow \mathbb{R}^{n\times n}$ be an invertible linear map that preserves the Frobenius inner ...
curiousperson's user avatar
3 votes
1 answer
82 views

Why $A(E)$ is open relative to $Y$ if $E$ is open?

Suppose $A$ is a linear transformation of $R^n$ into $R^m$. Let $Y=\{ Ax:x\in R^n \}$. I need to prove $A(E)$ is open relative to $Y$ for every open set $E \subset R^n$. My try is as follows: For ...
Winston's user avatar
  • 169
0 votes
0 answers
29 views

What affine transformation does the projective transformation correspond to?

In the projective space $P^3(\mathbb{K})$ with the frame $(S_0,S_1,S_2,S_3;E)$, consider a projective transformation as follows: $$\begin{cases} tx_0'=x_0\\ tx_1'=-x_1\\ tx_2'=-x_2\\ tx_3'=x_3.\end{...
Alex Nguyen's user avatar
0 votes
0 answers
27 views

Decoupling Linearly Coupled Wave Equations

I'm currently working numerically with wave equations and I was wondering if one can always decouple two wave equations, with potentials, which are linearly coupled. The system I'm talking about is ...
Afraxad's user avatar
2 votes
0 answers
69 views

Prove that $A$ is a linear operator mapping from $\ell^2$ to $\ell^2$. Determine $\|A\|$, $A^*$ and $\sigma(A)$.

For every $x = (x_n) \in \ell^2$, let $$ Ax = \left(x_1, \frac{x_2}{2}, x_3, \frac{x_4}{2^2}, x_5, \frac{x_6}{2^3}, \ldots \right). $$ Prove that $A$ is a linear operator mapping from $\ell^2$ to $\...
user avatar
0 votes
2 answers
42 views

Axler Theorem 5.18: $\text{null}(T)$ and $\text{range}(T)$ are invariant under $T$.

I am trying to understood Axler's proof of Theorem 5.18. It states that: if $T$ is a linear operator from $V$ to $V$ and $p$ is a polynomial with coefficients in the field $F$, then $\text{null} p(T)$ ...
Cardinality's user avatar
  • 1,279
0 votes
0 answers
140 views

Questions regarding the Hilbert space operator $x\mapsto \langle x, a \rangle a$

Let $H$ be a Hilbert space with $\dim H \geq 2$ and let $0 \neq a \in H$. Define the operator $A: H \to H$ by the rule $$ Ax = \langle x, a \rangle a \quad \text{for every } x \in H. $$ (a) Prove that ...
user avatar
1 vote
1 answer
51 views

For a given $k \in \mathbb{N}$, define the mapping $A: X \to X$ by the rule $ (Af)(x) = x^k f(x). $

Let $X = (C[-1,1], \|\cdot\|_\infty)$. For a given $k \in \mathbb{N}$, define the mapping $A: X \to X$ by the rule $ (Af)(x) = x^k f(x). $ (a) Prove that $A$ is a bounded linear operator. (b) ...
user avatar
1 vote
0 answers
38 views

Axler Theorem 5.17, part (b)

I am trying to understand the proof of part (b) of Theorem 5.17 in Axler's Linear Algebra Done Right. He cites part (a) in his proof of (b), so I've written out the full theorem statement below. $\...
Cardinality's user avatar
  • 1,279
0 votes
0 answers
59 views

If $ \phi: X \to X^* $ is an isometry, then $ X $ is a complete space.

I am wondering if the following statement might hold (as I wanted to use this in solving another problem): If $ \phi: X \to X^* $ is an isometry, then $ X $ is a complete space. I know that $ X^* $ is ...
user avatar
0 votes
0 answers
32 views

Proving that cylinder-to-sphere projection is area preserving.

I was going through the textbook Computer Graphics: Principles and Practice, and in the chapter for light, it talks about how the projection from a cylinder to sphere is area-preserving. I was trying ...
kre 1's user avatar
  • 1
0 votes
0 answers
45 views

Is the derivative at a point x of a smooth real-valued map linear?

I am currently reading "Differential Topology" by Victor Guillemin and Alan Pollack. They are in the process of explaining the preimage theorem in terms of a set of common zeroes (to show ...
Tosca's user avatar
  • 41
0 votes
1 answer
44 views

If $f(x) = f(x − a) + f(x − b) − f(x − a − b)$, is $f$ $\Bbb Q$-linear?

This question appeared in a test : A map $f: V → W$ between finite dimensional vector spaces over $\Bbb Q$ is a linear transformation if and only if $f(x) = f(x − a) + f(x − b) − f(x − a − b)$, for ...
Soumyadeep mandal's user avatar
0 votes
1 answer
35 views

Generalized Rotational Matrix for n-dimensional Euclidean Vector Spaces [duplicate]

$R_{ij}(\theta) := \begin{bmatrix}I_{i-1} & 0 & ... & ... & 0 \\ 0 & \cos(\theta) & 0 & -\sin(\theta) & 0 \\ 0 & 0 & I_{j-i-1} & 0 & 0 \\ 0 & \sin(\...
nameless___'s user avatar
1 vote
0 answers
68 views

Real-valued projection matrix on generalized eigenspace of a matrix with complex eigenvalues

Consider the Jordan decomposition $A = VJV^{-1}$. A projection matrix on the $i$-th generalized eigenspace of $A$ can be constructed as follows: construct $V_i$, whose columns are the columns of $V$ ...
user594147's user avatar
3 votes
1 answer
53 views

Is it possible to efficiently create a matrix M in which the elements are the sum of all possible "path-products" of matrix A?

I have a lower triangular matrix A where $a_{ij}=0, j\geq i$. I want to build a lower triangular matrix M where $m_{ij}$ is the sum of all the possible "path-products" from index $i$ to ...
kabolat's user avatar
  • 31
0 votes
1 answer
34 views

Baby Rudin Theorem 9.7b: how to verify $\|A-B\|$ has the properties of a metric?

Baby Rudin Theorem 9.7: where $L(R^n,R^m)$ is the set of all linear transformations of $R^n$ into $R^m$. I need to prove the distance $\|A-B\|$ has the properties of a metric: (a) $ \quad d(p,q)>...
Winston's user avatar
  • 169
1 vote
1 answer
55 views

Understanding of Rudin Definitions 9.6c: $|A \mathbf{x}| \le \lambda|\mathbf{x}|$ for all $\mathbf{x} \in \Bbb{R}^{n}$ $\implies$ $\|A\| \le \lambda$

Let $A$ be a linear transformation of $\Bbb{R}^n$ into $\Bbb{R}^m$. Define the norm $\|A\|$ of $A$ to be the sup of all numbers $|A \mathbf{x}|$, where $\mathbf{x}$ ranges over all vectors in $ \Bbb{...
Winston's user avatar
  • 169
-2 votes
1 answer
50 views

Prove that $A$ is a bounded automorphism of the vector space $X$. Also prove that $\sigma(A) = \{ \lambda \in \mathbb{C} \mid |\lambda| = 1 \}$.

Let $X = \{ f : \mathbb{R} \to \mathbb{C} \mid f \text{ is a bounded function} \}$. We know that $X$ is a Banach space equipped with the norm $\| f \| = \sup_{x \in \mathbb{R}} | f(x) |$. Let $c \in \...
user avatar
0 votes
1 answer
41 views

Prove that the sequence $(A_n(f))$ is convergent in the normed space $C[0,1]$. Prove that the sequence $(A_n)$ is not convergent in the operator norm.

Let $C[0,1]$ be a normed space equipped with the norm $\|\cdot\|_\infty$, and let for every $n \in \mathbb{N}$, the mapping $A_n$ be given by the prescription $ (A_n(f))(x) = \begin{cases} f(x), &...
user avatar
0 votes
1 answer
38 views

understanding the dimension theorem proof and analysing it

background : finished linear algebra 1 learning linear algebra 2, I am trying to reach a solid understanding of the theorems we learnt and prove them on my own ,while connecting them to previous ...
dareen's user avatar
  • 61
0 votes
1 answer
52 views

Prove that the canonical mapping from an infinite dimensional vector space to it's double dual is a one-to-one mapping. [closed]

What is the canonical correspondence from a vector space V to it's double dual $V^{**}.$ Prove that this correspondence is one-one.($V$ need not be finite dimensional) I tried solving the problem in ...
Thomas Finley's user avatar
0 votes
1 answer
95 views

How do I find maximal quotients of subspaces of the vector spaces $V_i$?

Say I have linear maps $V_1 \xrightarrow{h} V_2 \xleftarrow{g} V_3 \xrightarrow{f} V_4$. Assume $V_i$ are finite dimensional. (1) I want to find maximal subspaces $W_i$ of $V_i$ such that in the ...
J. Doe's user avatar
  • 752
1 vote
0 answers
54 views

Linear operator annihilates vector for every polynomial

I am working on this problem. I am studying for an exam. Let $T$ be a linear operator on a finite-dimensional vector space $V$. Show that there is a non-zero vector $v \in V$ such that for all $f \in \...
user123456's user avatar
0 votes
1 answer
60 views

Why is omitting a column that is a linear combination of other columns not considered an elementary operation in solving a system of linear equations? [closed]

Given a system of linear equations represented by the matrix equation $A\mathbf{x}=\mathbf{b}$, where $A$ is an $m \times n$ matrix. Why does removing a column from $A$ that is a linear combination ...
p0vi3's user avatar
  • 3
2 votes
2 answers
50 views

The left shift map and its image.

Let $X\subset{\mathbb{R}^{\infty}}$ be the space of convergent real sequences and let $L\colon{X}\to{X}$ be the operator such that $$L(x_1,x_2,x_3,\ldots)=(x_2,x_3,x_4,\ldots),\quad\forall\bar{x}=(x_1,...
Darkmaster's user avatar
0 votes
0 answers
42 views

Does permutation (row exchange) change the invertibility of a matrix? (a question on Gilbert Strang's Linear Algebra book)

On page 114 of Gilbert Strang's Linear Algebra (5th), it writes that If $A$ is invertible, a permutation $P$ will put its rows in the right order to factor $PA=LU$. There must be a full set of pivots ...
AStudent's user avatar

1
2 3 4 5
232