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4.868 linear-transformations questions.

In our lecture notes we're given some properties of Walsh-Transform, for a given mapping $f: F_2^n \mapsto F_2$ (from n-dimensional-Vectorspace over F2 to F2) The Walsh-Transform itself is given as $...

I have this linear transformation: $f(x,y,z) = (x+y+z, 2x+2y+2z, 3x+3y+3z)$ From $\mathbb{R}^3 \rightarrow \mathbb{R}^3$ I need to find the associated matrix with respect to the standard basis. How ...

Let $f: R^n\mapsto R^m$ a linear application. Is $f$ closed? Why? Is it open? Why?

I am missing something essential in my understanding of matrices as linear mappings (or matrices in general). To explain, I think I have a decent intuition for columnspace of a matrix. Basically, the ...

I was trying to solve a problem that asks me to show $p(Q^{-1}AQ) = Q^{-1}p(A)Q$ where $p(t)$ is an arbitrary polynomial $a_nt^n+...+a_1t + a_0$. I am wondering whether it is true that $(Q^{-1}AQ)^k = ...

Linear Algebra Problem. Please help. kindly teach me how to do it in detailed manner. Thanks for your time

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 ...

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 ...

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. ...

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 ...

Let $V$ be the vector space of differentiable functions $f : \mathbb{R \to C}$. Show that the transformation $T = \dfrac{\mathrm d}{\mathrm dt}$ defined as $\dfrac{\mathrm df}{\mathrm dt} = \dfrac{\...

Let $F:V\rightarrow U$ be a linear transformation. We have to show that the preimage of any subspace of $U$ is a subspace of $V$. My proof: Say $W$ is a subspace of $Im(F)$ and $T$ is a subspace of $...

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 ...

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 $f$...

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 ...

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 ...

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 $E$ ...

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 ...

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 ...

Let $\vec{v}_1,\ldots,\vec{v}_k$ be vectors in a vector space $V$ and $L\colon V \rightarrow W$ be a linear mapping. How would you prove that if $L(\vec{v}_1),\ldots,L(\vec{v}_k)$ spans $W$ that $\dim(...

I was asked to find orthogonal projection matrix in some basis, that projects onto $W: x_1+2x_2-3x_3=0$ and give the vectors for this basis I am using. So first I noticed that $\{(-2,1,0), (3,0,1)\}$ ...

Say $f:V\rightarrow U$ is a linear map with $\ker(f)=W$ and $f(v)=u$. Show that the set $v+W=\{v+w:w\in W\}$ is the preimage of $u$. My proof: $v \in W:f(v)=u$, $w\in W\implies f(w)=0$ $f(v+w)=u$ ...

Let $X$ a real or complex vector space and $F$ a subspace of $X$ with finite codimension (that is, there exists a finite dimensional subspace $E$ of $X$ such that $X=E+F$). Is it true that, for any ...

I have a matrix $\mathbf{A}$ such that $\mathbf{A'A}$ has ones along the diagonal and the off-diagonal elements are between $0$ and $1$. Do the transformations produced by $\mathbf{A}$ and $\mathbf{A'...

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 ...

I got this question: $V$ is a complex vector space with dim $V>1$. Show that for every bilinear transformation $\beta : V \times V \to \mathbb{C}$, there exists $0 \neq v \in V$ with $\beta (v,v) =...

I hope I translate this task properly. Tell me if something is unclear. I have to find transformation matrix for any isometric transformation at $\mathbb R^{3} $ which transform point (0, 0, -5) to (0,...

For a square matrix $M$ call any square matrix M' of the form $$\left(\begin{array}{cc} M & A\\ B & C \end{array}\right)$$ an extension of $M$. Does it follow that if $M$ is not invertible ...

given $V$ a vector space, let $\dim{V}=n$. $T\colon V\longrightarrow V$ is a linear transformtion with $\dim(\operatorname{Im}{T})=r$. Let $X$ be the set of all linear transformtion $F:V\...

Let $V$ be a vector space of all real valued functions from $(-1,1) \rightarrow \mathbb{R}$, $W_1 = \{f \in V \vert f(x) = f(-x)$ $\forall x \in (0,1)\}$, $W_2 = \{f \in V \vert f(-x) = -f(x)$ $\...

For n$\ne $ m let $ T_1 :R^n \to R^m $ and $ T_2:R^m\to R^n $ be linear transformations s.t $ T_1T_2 $ is bijective. Find rank of $ T_1$ and$ T_2$. I tried by fact that because $ T_1T_2 $ is ...

Let $T : l_2 \to l_2$ be a continuous linear operator. Let $B$ be the closed unit ball in $l_2$. How to show that the closure of $T(B)$ i.e. $\overline{T(B)}$ compact in $l_2$ where $T(x) = (x_1, ...

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 ...

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 ...

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 ...

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 ...

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 $\mathbb V(\oplus)$ ...

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(...

If $A=[1 -2 1; -1 1 2; 2 -1 3]$ be a matrix representation of a linear transformation T: P2(x) -> P2(x) with respect to bases {1-x,x-x^2, x+x^2} and {1,1+x,1+x^2}, then find T.

Let $V$ be an inner product space over $\mathbb{R}$, and $0\neq v \in V$ some vector. $T: V \rightarrow V$ is the operator defined as: $$ T(x) = x- \frac{2\left \langle v,x\right \rangle }{\left \...

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 ...

I have already found the eigen values for a particular linear transformation. Substituting with the eigenvalue $\lambda$ in a matrix I get: $C=\begin{bmatrix}-1 & -1 & -2\\-1 & -1 & 2\...

Let $f\colon V\rightarrow W$ be a linear transformation with $V,W$ $\mathbb{K}$-vector spaces and $U_{1},U_{2}$ subspaces of $W$ Show that: $f^{-1}(U_{1})+f^{-1}(U_{2})\subset f^{-1}(U_{1}+U_{2})$...

On the space of $2\times 2$ matrices, let $T$ be the transformation that transposes every matrix. Find the eigenvalues and "eigenmatrices" for $A^T =\lambda A$. By taking determinants on the left and ...

I'm studying symmetry operations and trying to show that the symmetry operations of a crystal* are closed, so they form a group. A friend of mine said that these operations commute but I can't justify ...

Let $T\colon V \to V$ be a linear endomorphism of a finite dimensional $k$-vector space $V$. Suppose that the minimal polynomial of $T$ is of the form $p(X)^m$, for an irreducible polynomial $p(X) \...

Suppose $m$ is a nonnegative integer, $z_1,\cdots, z_{m+1}$ are distinct elements of $\mathbb{F}$, and $w_1, \cdots, w_{m+1} \in \mathbb{F}$. Prove there exists a unique polynomial $p\in P_{m}(\mathbb{...

For example we have α, β belongs to $\Bbb R$, then how to show that the following system of linear equations will have infinite many solutions whenever β belongs to $[−\sqrt 2,\sqrt2]$? $$\left\{ \...

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