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Question 1

-1votes

0answers

36views

Proof of Linearly independece set [duplicate]

Let $L_S:F^n→F^m$ be a linear map which has null space $O$.Prove: if $H^1,\dots,H^k$ are linearly independent elements in $F^n$, then $L_S (H^1 ),\dots,L_S (H^k )$ are linearly independent elements ...

William Avila Aguilar

39

askedNov 24 at 16:08

Question 2

3votes

3answers

191views

The only area–preserving linear symmetry of a strictly convex body fixing a boundary point is the identity?

If $K\subset\mathbb R^2$ is strictly convex, $T\in SL(2,\mathbb R)$ is linear with $T(K)=K$, and $T$ fixes some boundary point $p\in\partial K$, must $T$ be the identity?Without strict convexity, we ...

hbghlyj

6,067

askedNov 21 at 5:59

Question 3

2votes

3answers

143views

Is there a similar way of proving this statement when $\mathbb{K}=\mathbb{F_2}$?

I am proving that $\mathrm{Z(GL}(V))$ is in $\mathbb{K}I_n$. $V$ is a finite dimensional vector space of dimension $n\geq2$ over a field $\mathbb{K}$. The problem I have is that this prove does not ...

amexan

41

askedNov 18 at 21:26

Question 4

0votes

1answer

46views

Can a 2D linear transformation have a finite set of invariant lines?

I've been trying to prove that if a 2D linear transformation has an invariant line through the origin then it must have an infinite set of invariant lines that are parallel to this line. I've searched ...

user1596992

13

askedNov 9 at 22:28

Question 5

1vote

1answer

172views

Find adjoint operator of $L(f) = f' + 3f$

Problem Setup: Let $V = \mathcal{P}_{1}$ with $\langle f,g \rangle = \int_{-1}^{1} f(x)\,g(x)\,\mathrm dx$ be the inner product space. Here $\mathcal{P}_{1}$ is the space of all linear polynomials. ...

Miranda

1,191

askedNov 1 at 22:07

Question 6

0votes

1answer

66views

Adjoint of a linear transformation

I am confused about the definition of the adjoint of a linear transformation, after looking at it in different sources and with different questions.(1) Some sources state is as: given $T:V\rightarrow ...

Maths Rahul

3,485

askedNov 1 at 9:53

Question 7

0votes

0answers

53views

Construction of a non-degenerate symmetric bilinear form from a given linear operator

T is a linear operator on a finite dimensional vector space V. Show that there exists a non-degenerate symmetric bilinear form B on V such that B(Tu,v)=B(u,Tv) for all u,v in V.I tried to use the ...

David

1

askedOct 30 at 14:51

Question 8

2votes

3answers

141views

How to show that a tripotent matrix (defined by the equation $A^3 = A$) is diagonalizable?

A tripotent matrix is a special case of $k$-idempotent matrix with $k = 3$. That is, a tripotent matrix is defined by the equation $A^3 = A$. (A trivial example is $A = I$.) The possible eigenvalues ...

Zach

2,948

askedOct 23 at 2:50

Question 9

7votes

2answers

151views

$f(A)$ is invertible $\iff$ $A$ is invertible. Then show that $\det f(A)$ = $c \cdot \det A$ for some $c$.

Let $f:M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})$ be a linear map satisfying $f(A)$ is invertible iff $A$ is invertible. Show that det $f(A)$ = $c$ det $A$ for some constant $c$.My idea is that let $...

Jdhejw

581

askedOct 19 at 5:30

Question 10

4votes

3answers

95views

Proving equivalence of $T$-invariance of $W$ and $T^{*}$-invariance of $W^{\perp}$

I'm working in my homework of linear algebra and there is this problem I'm strugling withLet $V$ be a finite-dimensional vector space with an inner product, and let $T$ be a linear operator on $V$. I'...

Jisbon

131

askedOct 15 at 6:14

Question 11

1vote

1answer

64views

Limit case of Bernstein's inequalities for Markov chain with spectral gap

Context:Let $\pi$ be a (potentially continuous) probability distribution.Let $\mathcal{L}^2(\pi)$ be the set of square-integrable function (real-valued) with respect to $\pi$, equipped with the ...

ylefay

178

askedOct 14 at 16:55

Question 12

1vote

1answer

57views

Functions of transpositions of symmetric group

The following is from Orthogonal Polynomials of Several Variables by Charles F Dunkl and Yuan Xu 2nd edition , Encyclopedia of math..and applications 155 page 320. Here i am assuming (i,j) means the ...

user158293

498

askedOct 12 at 10:48

Question 13

1vote

1answer

100views

Question about an expression of the norm $\|A\|_{\infty\rightarrow 1}$

Given a matrix $A$, the norm $\|A\|_{\infty\rightarrow 1}$ is $\max_{x:\|x\|_{\infty}=1}\|Ax\|_1$.A reference says that$$\|A\|_{\infty\rightarrow 1}=\max \sum_{i,j}A_{ij}c_id_j,$$where the maximum ...

Connor

2,504

askedOct 7 at 21:58

Question 14

0votes

0answers

54views

Linear operator of change of coordinate matrix - Hoffman Kunze sec 3.4

The question is related to section 3.4, page 92 from Hoffman and Kunze. It's a discussion near the end of the section.I'm able to deduce that $$[U]_B = P$$,My question is that for any $\alpha \in V$ ...

Ark

349

askedOct 3 at 21:06

Question 15

0votes

0answers

80views

Geometric Effect of Householder Reflector

We have the Householder reflector $F = \begin{bmatrix}-c & s\\s &c\end{bmatrix}$, where $c$ stands for $\cos{\theta}$, $s$ stands for $\sin{\theta}$ for some $\theta$. We want to know its ...

Miranda

1,191

askedSep 29 at 2:23

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Questions tagged [linear-transformations]

Proof of Linearly independece set [duplicate]

The only area–preserving linear symmetry of a strictly convex body fixing a boundary point is the identity?

Is there a similar way of proving this statement when $\mathbb{K}=\mathbb{F_2}$?

Can a 2D linear transformation have a finite set of invariant lines?

Find adjoint operator of $L(f) = f' + 3f$

Adjoint of a linear transformation

Construction of a non-degenerate symmetric bilinear form from a given linear operator

How to show that a tripotent matrix (defined by the equation $A^3 = A$) is diagonalizable?

$f(A)$ is invertible $\iff$ $A$ is invertible. Then show that $\det f(A)$ = $c \cdot \det A$ for some $c$.

Proving equivalence of $T$-invariance of $W$ and $T^{*}$-invariance of $W^{\perp}$

Limit case of Bernstein's inequalities for Markov chain with spectral gap

Functions of transpositions of symmetric group

Question about an expression of the norm $\|A\|_{\infty\rightarrow 1}$

Linear operator of change of coordinate matrix - Hoffman Kunze sec 3.4

Geometric Effect of Householder Reflector

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