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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"Every linear mapping on a finite dimensional space is continuous"

From Wiki Every linear function on a finite-dimensional space is continuous. I was wondering what the domain and codomain of such linear function are? Are they any two topological vector spaces (not ...
Tim's user avatar
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The range of $T^*$ is the orthogonal complement of $\ker(T)$

How can I prove that, if $V$ is a finite-dimensional vector space with inner product and $T$ a linear operator in $V$, then the range of $T^*$ is the orthogonal complement of the null space of $T$? I ...
user62182's user avatar
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82 votes
8 answers
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Equivalent Definitions of the Operator Norm

How do you prove that these four definitions of the operator norm are equivalent? $$\begin{align*} \lVert A\rVert_{\mathrm{op}} &= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for ...
KiaSure's user avatar
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6 votes
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Determining possible minimal polynomials for a rank one linear operator

I have come across a question about determining possible minimal polynomials for a rank one linear operator and I am wondering if I am using the correct proof method. I think that the facts needed to ...
user7980's user avatar
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18 votes
2 answers
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Show that the operator norm is submultiplicative

We had in our lecture on numerical analysis the following: Let $\mathrm{Lin}(X,Y)$ be the set of all linear maps $X\rightarrow Y$. Let $A\in\mathrm{Lin}(\mathbb R^l,\mathbb R^n)$ and $B\in\mathrm{Lin}(...
user58679's user avatar
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12 votes
3 answers
4k views

Orthogonal Projection

Seems like I still don't get it, I think I am missing something important. Let $V$ be an $n$ dimensional inner product space ($n \geq 1$), and $T\colon\mathbf{V}\to\mathbf{V}$ be a linear ...
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45 votes
3 answers
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When can two linear operators on a finite-dimensional space be simultaneously Jordanized?

IN a comment to Qiaochu's answer here it is mentioned that two commuting matrices can be simultaneously Jordanized (sorry that this sounds less appealing then "diagonalized" :P ), i.e. can be brought ...
Mark's user avatar
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48 votes
6 answers
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Reflection across a line?

The linear transformation matrix for a reflection across the line $y = mx$ is: $$\frac{1}{1 + m^2}\begin{pmatrix}1-m^2&2m\\2m&m^2-1\end{pmatrix} $$ My professor gave us the formula above with ...
dsd's user avatar
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10 votes
1 answer
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Basis for $\mathbb R$ over $\mathbb Q$

Give me some examples of basis for $\mathbb R$ (as vector space over field $\mathbb F=\mathbb Q$). Thanks.
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6 votes
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A linear transform of a closed set is closed

A linear transform of a closed set $E\subset \mathbb{R}^d \to \mathbb{R}^d$ is closed. I have seen a lot of similar questions here, but none of them exactly addresses the issue. Please if you find it ...
Susan_Math123's user avatar
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suppose $|a|<1$, show that $\frac{z-a}{1-\overline{a}z}$ is a mobius transformation that sends $B(0,1)$ to itself.

Suppose $|a|<1$, show that $f(x) = \frac{z-a}{1-\overline{a}z}$ is a mobius transformation that sends $B(0,1)$ to itself. To make such a mobius transformation i tried to send 3 points on the edge ...
Kees Til's user avatar
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39 votes
3 answers
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Is a map that preserves lines and fixes the origin necessarily linear?

Let $V$ and $W$ be vector spaces over a field $\mathbb{F}$ with $\text{dim }V \ge 2$. A line is a set of the form $\{ \mathbf{u} + t\mathbf{v} : t \in \mathbb{F} \}$. A map $f: V \to W$ preserves ...
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29 votes
1 answer
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Is there any simple set of properties which uniquely characterizes differentiation?

The transformation of differentiation is a linear operator over $C^\infty(\mathbb{R}),$ the vector space of smooth functions over $\mathbb{R}.$ Is there any simple set of properties that uniquely ...
mathlander's user avatar
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21 votes
4 answers
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Find the standard matrix for a linear transformation

If $T: \Bbb R^3→ \Bbb R^3$ is a linear transformation such that: $$ T \Bigg (\begin{bmatrix}-2 \\ 3 \\ -4 \\ \end{bmatrix} \Bigg) = \begin{bmatrix} 5\\ 3 \\ 14 \\ \end{bmatrix}$$ $$T \Bigg (\begin{...
user60912's user avatar
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15 votes
2 answers
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What shape do we get when we shear an ellipse? And more generally, do affine transformations always map conic sections to conic sections?

What shape do we get when we shear an ellipse? Is it another ellipse (or circle in special cases)? Or is it some other shape which isn’t a conic section? I was under the impression that applying any ...
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12 votes
1 answer
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Do full rank matrices in $\mathbb Z^{d\times d}$ preserve integrals of functions on the torus?

Let $\mathbb T:=[0,1]/(0\sim 1)$. Its easy to see that if $f:\mathbb T\to \mathbb R$ and $k\in\mathbb Z$ is not zero, then $f_k(x):=f(kx)$ (1) defines a map $ f_k:\mathbb T\to \mathbb R$, and $$ \...
Calvin Khor's user avatar
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9 votes
2 answers
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Proving that a linear isometry on $\mathbb{R}^{n}$ is an orthogonal matrix

I wish to prove that if $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is defined by $T(v)=Av$ (where $A\in M_{n}(\mathbb{R})$) is an isometry then $A$ is an orthogonal matrix. I am familiar with many ...
Belgi's user avatar
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8 votes
1 answer
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''Linear'' transformations between vector spaces over different fields

Let $\mathbf{V}(\mathbb{K}_1,V)$ and $\mathbf{W}(\mathbb{K}_2,W)$ be two vector spaces over different fields (as an example, $\mathbb{K}_1=\mathbb{C}$ and $\mathbb{K}_2=\mathbb{R}$). Can we generalize ...
Emilio Novati's user avatar
4 votes
1 answer
4k 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{...
mathgenius's user avatar
4 votes
3 answers
930 views

If $ A^3=A$ prove that $Ker\left(A-I\right)+Im\left(A-I\right)=V$

If $ A^3=A$ prove that $Ker\left(A-I\right)+Im\left(A-I\right)=V$ I am not sure how to approach this problem, but first things first if we have $A^3=A$ that is $A^2=I$, what does that tell me (what ...
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3 votes
3 answers
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If $T\colon \mathbb R^n \to \mathbb R^n $ linear and $T^2 = kT$ [closed]

It is given that $T$ is a linear transformation from $\mathbb R^n$ to $\mathbb R^n$ such that $T^2 = k T $ for some $k\in \mathbb R$. Then, one or more of the options are true $\|T(x)\| = |k| \|x\...
preeti's user avatar
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2 votes
0 answers
468 views

Characterisation of inner products preserved by an automorphism

Let $V$ be a finite dimensional vector space. Let us call an automorphism $T:V\rightarrow V$ admissible if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry. (You can ...
Asaf Shachar's user avatar
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1 vote
1 answer
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If $(Tx \mathbin{|} x) = 0$ for all $x$ then $T = 0$

Let $T$ be a linear operator on an inner product vector space $V$. I'd like to prove that if $$(Tx|x)=0 \quad \forall x \in V$$ then $T$ is the null operator. I can't figure out this proof using ...
omega-stable's user avatar
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1 vote
2 answers
935 views

Any linear subspace has measure zero [duplicate]

Definition Let $A$ be a subset of $\Bbb R^n$. We say $A$ has measure zero in $\Bbb R^n$ iffor every $\epsilon>0$, there is a covering $Q_1,\,Q_2,...$ of $A$ by countably many rectangles such that $$...
Antonio Maria Di Mauro's user avatar
26 votes
10 answers
29k views

Is matrix transpose a linear transformation?

This was the question posed to me. Does there exist a matrix $A$ for which $AM$ = $M^T$ for every $M$. The answer to this is obviously no as I can vary the dimension of $M$. But now this lead me to ...
avz2611's user avatar
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19 votes
3 answers
37k views

How do I find a dual basis given the following basis?

$V = \Bbb{R}^3$ and has basis $\mathcal{B} = \{\vec{e_1}-\vec{e_2},\vec{e_1}+\vec{e_2},\vec{e_3}\}$ How do I find the dual basis? This is not homework, but an example that I am struggling to grasp. ...
aidandeno's user avatar
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18 votes
1 answer
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The rank of a linear transformation/matrix

I'm terribly confused on the concept of "rank of a linear transformation". My book keeps using it, but it doesn't clarify what it means (or at least I haven't been able to find it). Is it the same as ...
Nico Bellic's user avatar
9 votes
2 answers
7k views

Proving a linear transformation is unique

In Axler's Linear Algebra Done Right, there is this theorem. (3.5) Suppose $v_1. . . v_n $ is a basis of $V$ and $w_ , . . . w_n \in W$. Then there exists a unique linear map $T: V \rightarrow ...
Peter_Pan's user avatar
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7 votes
2 answers
12k views

Show that the Area of image = Area of object $\cdot |\det(T)|$? Where $T$ is a linear transformation from $R^2 \rightarrow R^2$

Prove that the area of an image in $2d$ cartesian coordinates is equal to the determinant of the linear transformation times the area of the initial shape. I've tried to formulate general expression ...
jg mr chapb's user avatar
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4 votes
2 answers
392 views

Definitions of "linearity" across branches of mathematics or levels of math education

Linearity is a ubiquitous concept in mathematics; however, each branch of mathematics appears to have its own definition of what a linear map (function, functional, functor, transformation, form, ...
Björn Friedrich's user avatar
3 votes
1 answer
2k views

Active and passive transformations in Linear Algebra

I am trying to understand what each transformation means and what their differences are but many books that don't state which transformation they are referring to make it a bit confusing to understand ...
TheQuantumMan's user avatar
3 votes
2 answers
451 views

If $A$ and $B$ are linear transformations on a finite-dimensional inner product space, and if $\textbf{0} \leq A \leq B$, then det $A \leq$ det $B$.

Exercise 12 from SEC. 82 of Finite-Dimensional Vector Spaces - 2nd Edition by Paul R. Halmos. If $A$ and $B$ are linear transformations on a finite-dimensional inner product space, and if $\textbf{0} ...
Andreo's user avatar
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1 vote
3 answers
5k views

The eigenvectors of the transpose operator

Define $T : M_{n×n}(\mathbb{R}) → M_{n×n}(\mathbb{R})$ by $T(A) := A^t$. I know that the corresponding eigenvalues are $+1$ and $-1$, but I'm not sure how to find the eigenvectors of this ...
user avatar
40 votes
6 answers
23k views

The logarithm is non-linear! Or isn't it?

The logarithm is non-linear Almost unexceptionally, I hear people say that the logarithm was a non-linear function. If asked to prove this, they often do something like this: We have $$ \ln(x + ...
Björn Friedrich's user avatar
22 votes
2 answers
8k views

A real function which is additive but not homogenous

From the theory of linear mappings, we know linear maps over a vector space satisfy two properties: Additivity: $$f(v+w)=f(v)+f(w)$$ Homogeneity: $$f(\alpha v)=\alpha f(v)$$ which $\alpha\in \...
CLAUDE's user avatar
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16 votes
2 answers
6k views

Minimal polynomial of restriction to invariant subspace divides minimal polynomial

I'm trying to prove this: $T : V \to V$ linear transformation. $W$ subspace of $V$. If $W$ is $T$-invariant then the minimal polynomial for the restriction operator $T|_W$ divides the minimal ...
Nikki's user avatar
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9 votes
4 answers
3k views

Under what conditions is a linear automorphism an isometry of some inner product?

Assume $V$ is a finite-dimensional vector space over $\mathbb{R}$, and that $T: V \to V$ is a (linear) isomorphism. When is it possible to construct an inner product on $V$ making $T$ an ...
Asaf Shachar's user avatar
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8 votes
2 answers
741 views

linear operator $f(X) = AXB$

What are the eigenvalues of the linear operator in vector space $M_n(\mathbb R)$ $$ f(X) = AXA^T $$ and $$ f(X) = AXA^{-1} $$ when eigenvalues of $A$ are $ \lambda_1, \lambda_2, ..., \lambda_n $? ...
J. Doe's user avatar
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6 votes
1 answer
5k views

Expression of rotation matrix from two vectors

What is the matrix expression of the rotation matrix in 3D which turns a vector $\vec{a}$ into a vector $\vec{b}$, with both vectors given by their coordinates? ($\vec{a} = (a_x, a_y, a_z)$ and $\vec{...
tmlen's user avatar
  • 338
6 votes
3 answers
2k views

If $T^m$ is diagonalizable for a $m\in\mathbb N$, then $T$ is diagonalizable.

Suppose that $V$ is a finite dimensional $\mathbb C$-vector space, and suppose that $T:V\rightarrow V$ is injective. If there is a $m\in\mathbb N$ such $T^m$ is diagonalizable, then $T$ is ...
ett's user avatar
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6 votes
5 answers
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difference between linear transformation and its matrix representation .

I can't understand this: Given a matrix T =$\small \{T_{ij}\} \in \mathbb M_{mn}$,define a transformation $\small T:\mathbb R^n \rightarrow \mathbb R^m$ as follows: If $~~\small T(x_1,\ldots \...
spectraa's user avatar
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5 votes
1 answer
2k views

If $0$ is the only eigenvalue of a linear operator, is the operator nilpotent

In a finite dimensional vector space, if $0$ is an eigenvalue and the only eigenvalue of a linear operator, is that operator nilpotent? There is this post which shows the other direction. Prove that ...
user avatar
4 votes
2 answers
645 views

$T$ has no eigen-values

I would like to solve the following exercise: Let $V$ be the space of continuous real-valued functions on the real line. Let $T$ be the linear map on $V$ defined by $$ (Tf)(x) = \int\limits_{0}^{x}f(...
Amit's user avatar
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4 votes
1 answer
855 views

Proof that $\|fx\| \leq \|f\|\cdot\|x\|$

From the wiki article on the dual of a norm: $X$ and $Y$ are normed spaces, and we associate with each $f\in L(X,Y)$ (the space of bounded linear operators from $X$ to $Y$) the number $$\|f\| = \sup\{...
man_in_green_shirt's user avatar
3 votes
3 answers
1k views

Let $T,S$ be linear transformations, $T:\mathbb R^4 \rightarrow \mathbb R^4$, such that $T^3+3T^2=4I, S=T^4+3T^3-4I$. Comment on S.

Let $T,S$ be linear transformations, $T:\mathbb R^4 \rightarrow \mathbb R^4$, such that $T^3+3T^2=4I, S=T^4+3T^3-4I$. Then S is: one-one but not onto onto but not one one invertible non-invertible (...
Diya's user avatar
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3 votes
2 answers
2k views

Dual spaces and dual basis [closed]

If V is a finite dimensional vector space over the field F with dual space V* = Hom(V,F) . How to prove every ordered basis for V* is the dual basis for some basis for V ?
Shona's user avatar
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1 vote
1 answer
107 views

Does this result require $A$ to be convex?

Let $X$ be a vector space, $Y$ a subspace of $X$, and $A \subset X$. Then algebraic interior of $A$ with respect to $Y$ is defined as $$ \operatorname{aint}_{Y} (A) := \{x \in X \mid \forall y\in Y, ...
Analyst's user avatar
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1 vote
1 answer
1k views

Matrix in canonical form of an orthogonal transformation

Let: $$A = \frac 12 \begin{pmatrix} 1 & -1 & -1 &-1 \\ 1 & 1 & 1 &-1 \\ 1 & -1 & 1 & 1\\ 1 &1 &-1 & 1\end{pmatrix} $$ Prove there exists an ...
B. David's user avatar
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0 votes
2 answers
6k views

Let $S:U\rightarrow V, \ T:V\rightarrow W$ and if $S$ and $T$ are both injective/surjective, is $T\circ S$ injective/surjective?

For injective side, in my opinion, we can find two square matrices $A, B$ with each injective but $AB$ has a zero row ($\dim\ker(AB)\ne 0$), so $TS$ is not injective. But I don't know how to find such ...
CoolKid's user avatar
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0 votes
1 answer
1k views

Double dual mappings

$\newcommand{\Hom}{\operatorname{Hom}}$Let $U,V$ be vector spaces. Denote the dual space of $U$ with $U^\intercal$ and the dual mapping of a linear map $\Phi$ be $\Phi^\intercal$. Define the double ...
Henricus V.'s user avatar
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