# linear-transformations's questions - English 1answer

4.868 linear-transformations questions.

### 1 For $\mathbb{S}$ subspace of $\mathbb{V}$, prove $L:\mathbb{V}\to\mathbb{V}$ such that Ker$(L)=\mathbb{S}$

Let $\{\vec{v}_1,\ldots,\vec{v}_k\}$ be a basis for a subspace $\mathbb{S}$ of an $n$-dimensional vector space $\mathbb{V}$. Prove that there exists a linear mapping $L:\mathbb{V}\to\mathbb{V}$ such ...

### Is my reasoning on Transformation Matrices right?

Let $A=\{v_1,\cdots,v_n\}$ and $B=\{w_1,\cdots,w_n\}$ be two basis of $V$. Then we define $[v]_A = \{\alpha_1,\cdots,\alpha_n\}^T$ as the coordinate tuple of $v$ respect to $A$. We want to obtain the ...

### 3 How many “elementary” isometries are needed to generate any isometry of an n-dimensional Euclidean space?

I use here the term Euclidean space in the rigorous sense of an affine space over $\mathbb{R}^n$, equipped with the Euclidean inner product (see here). Let $\mathbb{E}^n$ be the Euclidean $n-$space. ...

### Why is L(z)=Az+B a linear transformation in Complex Analysis?

Recently, I am learning complex analysis using " Complex Analysis for mathematics and engineering" by John H. Mathews and Russell W. Howell. In Chapter 2, it says that "...because a linear ...

### 2 Orthogonal group is subgroup

How to show that Orthogonal group is subgroup of general linear group GL(V) where V is vector space? Since general linear group GL(V) group of all linear transformation which are bijective i.e ...

### -1 Fourier transform of $f\circ A$ where $A$ linear and $f\in L^1(\mathbb{R}^n)$

Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a linear map and $f\in L^1(\mathbb{R}^n)$. I would like to express the fourier transform of $f_A:= f\circ A$ in terms of the Fourier transform $\hat{f}$ of $f$...

### 1 Linear transform of exchangeable random vector

Say we have a finite random vector of exchangeable variables $X =(X_{1},X_{2},\dots,X_{m})$, i.e., $(X_{1},X_{2},\dots,X_{m}) \stackrel{d}{=} (X_{\pi(1)},X_{\pi(2)},\dots,X_{\pi(m)})$ for any finite ...

### Linear algebra curiosity [duplicate]

Is it true that if two linear functional from a vector space to the respective field have the same kernel, we can say that one is a multiple of the other? It is not so difficult to prove this when we ...

### Second adjoint into a subspace

Let $T$ be a continuous linear map from between metrizable locally convex spaces $E$ and $F$. Let $H$ be a closed subspace of $F$ such that $TE\subset H$. Then we can also view $T$ as a map from $E$ ...

### 3 Does there exist 2 matricies, such that they can be used to transpose any n by n matrix?

5 answers, 101 views linear-algebra linear-transformations
Ideally $\exists A, B$ to be able to transpose matrix $X \; \forall X \in M_{n\times n}$ by matrix multiplication. (Even more ideal is if there is only one matrix, $A$ that can transposes $X$ as ...

### Diagonalizing symmetric matrix [duplicate]

Can a symmetric matrix with all real entries in the matrix be diagonalized by special orthogonal matrix or a special unitary matrix ? Since it is symmetric I guess orthogonal transformation can ...

### For a nilpotent mapping $N$, why is it true that there exists a non-zero vector $x$ that $N^k(x)$ is zero but $N^{k-1}(x)$ isn't?

I was reading "A Course in Linear Algebra" by David Damiano and John Little. The third paragraph of chapter 6.2 confused me. Let me quote: Let $\text{dim}(V) = n$, and let $N: V \rightarrow V$ be a ...

### -1 How to prove the invertibility of the following opeator? [closed]

For $x = (x_n) \in l_2$, define $$T(x) = (0, x_1, x_2, \dots)\ \ \text{and} \ \ S(x) = (x_2, x_3, \dots)$$ Which of the following statements are true? If $A \colon l_2 \to l_2$ is a continuous linear ...

### 1 verification for axis symmetry in 3d space a line passing through two points

In an orthonormal coordinate system $K=O \overrightarrow{e_1}\overrightarrow{e_2} \overrightarrow{e_3}$ I have to find an analytic representation of an axis symmetry with an axis g represented by two ...

### 5 Annihilator of a vector space $V$ is the zero subspace of $V^*$

I am reading Hoffman and Kunze's Linear Algebra and in Section 3.5, page 101, they define the annihilator of a subset as follows: Definition. If $V$ is a vector space over the field $F$ and $S$ is ...

### Find the image of this Linear Transformation

The Question: Let $V$ be a finite-dimensional complex inner product space. Let $W$ be a subspace of $V$ with basis $\{ e_1,\dots,e_k\}$. Let $T:V \rightarrow W$ be a linear transformation defined ...

### 1 Prove this homomorphism gives an equivalent definition of Linear Space

Let $\mathbb V(\oplus)$ be an abelian group and $\mathbb K(+,\cdot)$ a field; of course $\mathbb K(\cdot)$ is the group under product. We represent the group of automorphisms of $\mathbb V(\oplus)$ ...

### 1 given $T:V\longrightarrow V$ a linear map ,$U=\{S\in Hom_{\mathbb{F}}(V,V):\ S\circ T=0\}$, prove: $U$ is a subspace of $Hom_{\mathbb{F}}(V,V)$

let $T:V\longrightarrow V$ be a linear map (transformation). let $$U=\{S\in Hom_{\mathbb{F}}(V,V):\ S\circ T=0\}$$ prove that $U$ is a subspace of $Hom_{\mathbb{F}}(V,V)$. someone send the ...