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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1answer
39 views

Definition of eigen space.

I am studying linear algebra and got confused in defining eigen space corresponding to eigen value.The thing wondering me is the same thing defined in two different books in different manners.let $\...
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1answer
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Matrix of the derivative operation $D$ with different basis

Suppose $V=\{p(x) : p(x) \text{ is a polynomial so that its degree is less or equal to 2}\}$ and $W=\{p(x) : p(x) \text{ is a polynomial so that its degree is less or equal to 1}\}$ are two vector ...
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2answers
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Define a linear transformation $T$, so that the null space is $z$-axis, and the range is the plane $x+y+z=0$

As stated in the title, it is requested to define a linear transformation $T:\Bbb R^3 \to \Bbb R^3$ such that the null space of $T$ is the $z$-axis, and the range of $T$ is the plane: $x+y+z=0$ I ...
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1answer
33 views

Let $V$ be a finite dimensional vector space and $W$ is proper subspace of $V$. Then show that Span of $(V/W)= V$.

Let $V$ be a finite dimensional vector space and $W$ is proper subspace of $V$. Then show that Span of $(V/W)= V$. I am trying to show that $V/W$ contain a basis of $V$ but How to proceed ? any ...
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1answer
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Find the value of $x$ such that the matrix represents a projection

Given the operator $P: \Bbb {R^3} \to \Bbb {R^3}$, whose matrix in the standard base is \begin{equation*} P_{3,3} = \begin{pmatrix} 1/2 & -1/2 & 1/2 \\ -1 & 0 & 1 \\ -1/2 & -1/2 &...
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2answers
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$\dim V \geq \dim U$, then there exists a linear injection from $U$ to $V$.

Let $U$ be a $K$ vector space with $\dim U=n$. I have to show that for all $r\in \mathbb{N}$ with $n\leq r$, there is a $K$ vector space $V$ with $\dim V=r$ and a linear injection $U\...
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0answers
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Linear map in different basis [closed]

So I have linear mapping given as: $$A(x,y,z)=(x+y-z,y,z)$$ and I have to find all basis where A is written as a matrix: $$A_B^B=\begin{pmatrix} 1 & 1 & 0\\ 0 & 1 & 0\\ 0 & 0 & ...
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1answer
23 views

If I have a matrix in RREF is there a quick way of pulling out vectors which are not in the span of the row vectors of A?

Say I am looking for the kernel of a linear function $\mathcal{l}_A$: $\mathbb{R}^n $->$\mathbb{R}^m$ given by the $n \times m$ matrix A. Then say A is row-equivalent to some matrix B in RREF. Is ...
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2answers
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X subset of space vector V, $Tv=Sv, v \in X$ then T=S

Given two linear transformations:$$S,T: V \to W$$ such that $X\subseteq V$, and it is true that $S(v)=T(v), \forall v\in X$, and the exercise requests to prove that $S=T$. I have started by writing ...
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1answer
51 views

Confused about the relation between linear transformations, matrices and basis vectors

I was watching 3blue1brown's video series on linear algebra. My understanding till now is :- A linear transformation takes in a vector and outputs another vector. The above statement is equivalent to ...
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0answers
19 views

incluson between kernels for two linearly dependent operators. [closed]

Let $X$ be a Banach space. Let $A$ and $B$ be two bounded operators in $B(X)$, assume that $Ax$ and $Bx$ are linearly dependent for each $x \in X$ but $A$ and $B$ are linearly independent. How we can ...
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30 views

When does a linear projection of a set of vectors preserve the span of the vectors?

Sorry if this question has been asked before, but I can't find it. Suppose I have a set of vectors with some span. If I apply the same linear projection $P$ to each vector in the set, the ...
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1answer
25 views

Linear Algebra: an example of a linear operator [closed]

Is there a linear operator on $\mathbb{C}^4$ that is not diagonalizable? Can anyone give me an example please.
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1answer
26 views

Projection on $U$ along $W$ with $U,W$ contained in $V=U\oplus W$

Assume $V=U\oplus W$ is a direct sum where $U$ and $W$ are subspaces of the vector space $V$. Then we can define a linear map $$E(v)=u,\\ with \qquad v=u+w\in V,\qquad u\in U,\ w\in W$$ Called the ...
2
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1answer
57 views

Riesz Representation Theorem geometric intuition

We just learned in our linear algebra class about the Riesz Representation Theorem, which states that if $V$ is finite-dimensional and $f$ is a linear functional on $V$, then there is a unique vector $...
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1answer
29 views

check if vectors are linearly independent?

Heyo, I'm just wondering if I'm correct in assuming that the following three vectors in four dimensional space are linearly dependent.. I simplified them using elemental row operations and ended up ...
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1answer
26 views

Unique differentiable linear operator mapping $\mathbb{R} \rightarrow \mathbb{R}^N$?

I am trying to find a mapping $\phi$ such that: $\phi$ uniquely maps $\mathbb{R}$ to the subspace of $\mathbb{R}^N$ (for bounded $N$), where every dimension of the vector is bounded between $-1$ and $...
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3answers
18 views

Matrix of the differentiation operation [duplicate]

Exercise: Find the matrix of the derivative operation $D$ related to the base $\{1, t, t^2,..., t^n\}$ $$D: \mathcal P_{n} \to \mathcal P_{n}$$ I found a possible solution to this exercise, given that ...
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1answer
34 views

Problem 1, Exercise 3.4, Linear Algebra, Hoffman and Kunze

The fundamental question is this: Let $V$ be a n-dimensional F-vector space. Is the matrix $A \in F^{n \times n}$ of $T \in L(V,V)$ relative to $\mathcal{B}$ is unique upto row-equivalence or not. ...
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0answers
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Conditions for existence of a Moore-Penrose inverse

Let $A$ be a (noncommutative) $*$-algebra, and think of $A$ like it were endomorphisms on a finite dimensional vector space. A Moore-Penrose inverse for $a \in A$ is $b$ such that $bab = b$. $aba = ...
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1answer
26 views

Is there a notion of maps that can “expand” spaces “linearly”?

For linear transformations, the dimension of the image is at most the dimension of the domain. More generally, given vectors $v_1, ..., v_n$ in the domain, the vectors $Tv_1, ..., Tv_n$ span the image....
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1answer
45 views

Prove that $U$ and $V$ are subspaces of $\mathbb{R}^3$ [closed]

Given $U = \{ (x, y, z) \in\mathbb{R}^3 | x-3y+z = 0 \}$ and $V = \{ (x, y, z) \in\mathbb{R}^3 | x+2z = 0 \}$, prove that $U$ and $V$ are subspaces of $\mathbb{R}^3$. I'm new to Linear Algebra so, I'...
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3answers
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Show that the projection is $\in End(V)$

Consider a direct sum $V=U\oplus W$ where $U,W \subseteq V$. In my lecture notes I have been given the definition of the projection on U along W: $E\in End(V), E(v)=u$ with $v\in V$ and $u\in U$. I ...
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1answer
27 views

Basis of annihilator of subspace in $\mathbb{R}^3$

I want to find a basis for the annihilator, $U^{\circ}$, of the subspace $U:=Span((1,0,0))\subseteq \mathbb{R}^3$. As far as I have understood, in $\mathbb{R}^n$ the linear functionals can be seen as ...
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0answers
17 views

Is there a term for a linear transformation that maps a standard basis element to a scalar multiplication of a standard basis element?

I'm studying quantum computing in purely algebraic sense, and I've come up with a question: Is it possible to decide whether the output of a quantum logic gate is always non-random, when fed non-...
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2answers
30 views

Show that every $u\in V$ is a sum of eigenvectors of P, i.e. $u=v+w$ where $u$ and $v$ are eigenvectors of P

Let $V$ be a vector space and $P:V\rightarrow V$ is a linear transformation such that $P \circ P=P. $ Show that every $u\in V$ is a sum of eigenvectors of P, i.e. $u=v+w$ where $v$ and $w$ are ...
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1answer
15 views

Preservation of linear separability under linear transformations

Let $a\in\mathbb R^n$ be a given vector. I separate vectors $x\in\mathbb R^n$ into two sets: $S^+ = \{x: a^Tx>0\}$ and $S^- = \{x: a^Tx \le 0\}$. Now I consider a linear transformation (matrix) $T:...
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0answers
28 views

Union of images under arbitrary linear maps with boundness assumptions

Let $\mathcal{A}\subset B(\mathbb{R}^n,\mathbb{R}^n)$ be a bounded family of linear maps (matrices). Now let $X_0 = \{ x\in\mathbb{R}^n ~\vert~ \lVert x \rVert \leq 1 \}$, where $\lVert \cdot \rVert$ ...
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2answers
37 views

Linear transformation of matrix [closed]

This is in relation to a question that was taken down. It was regarding a matrix of the form $$\begin{pmatrix} a&b \\ c&d \end{pmatrix}$$ Which after some transformation $X$ then comes in the ...
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1answer
13 views

How to find Skew Projection Operator onto Plane parallel to some vector?

I was trying to solve previous year question paper of competitive exam In that I observed some strange question which I have not encountered before. They had given one equation of plane and told to ...
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1answer
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For any two vectors $v_1$ and $v_2$, $\langle v_1, v_2 \rangle = \langle T (v_1), T (v_2) \rangle$ if $T$ is norm-preserving.

A linear transformation $T : V \to V$ is said to be norm-preserving for a given inner product if for all vectors $v$ in $V$ the following is true: $\langle v,v \rangle = \langle T(v),T(v) \rangle$. ...
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0answers
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How to find A such that $Ab$ is in the same direction of an eigenvector of $ACA^T$

Suppose $\mathbf{b}=[b_1,b_2]'$ is $2\times 1$ and $\mathbf{C}$ is a full-rank $2\times 2$ matrix which both are real and given. Now, consider the problem of finding a $2\times 2$ matrix $\mathbf{A}$ ...
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0answers
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Prove that it's a Linear Functional (Kronecker delta)

Suppose that $dim(V) = n$ and that $B = \{e_1,...,e_n \}$ is a basis for $V$. Then we define $B^{*} = \{e^1,...,e^n \}$ such that: \begin{equation} e^{i}(e_{j}) = \delta_{j}^{i} = \begin{cases}1, ...
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1answer
48 views

Expectation of product of three or more dependent random variables

Given three random variables $X,Y,Z$, what is the formula of expectation of the product of three random variables? $$\mathbb{E}[XYZ]=?$$ Will it be less complex if we assume $X, Y$ and $Z$ as ...
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2answers
43 views

How do I show if the following is a linear transformation? [closed]

I'm learning about the concept of linear transformation and I am having some trouble applying it. I know that for T to be a linear transformation, then $T(x+w) = T(x) + T(w)$ and $T(rx) = rT(x)$ Now,...
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0answers
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Decomposition of vector space to produce nilpotent and invertible transformations [duplicate]

Let $V$ be a finite-dimensional vector space over a field $K$. Let $T$ be a linear operator on $V$. Prove that there exists a unique sum $V=V_{0}+V_{1}$ such $T(V_{0}) \subseteq V_{0}$, $T(V_{1}) \...
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0answers
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Coordinate transformation for antenna pointing help

I may have bitten off more than I can chew with some coordinate transformation math. As a background, I'm working on a project where we need to mechanically point an antenna to a receiving tower. The ...
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0answers
21 views

Dual basis vectors of a normed space are bounded.

Let $V$ be a finite dimensional normed vector space over a field $\mathbb{F}$ with basis $\{e_1,...,e_n\}$, we know that for all $x\in V$ there are unique scalars $a_1(x),..,a_n(x)$ s.t $x=\sum_{i=1}^...
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0answers
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Dimension of the Annihilator and the dual of the quotient space.

Let $W\subset V$ a vector subspace(not necessarily finite). I've been tryin to prove that the $dim(Ann(W))=dim((V/W)^*)$. Actually, the two spaces are isomorphic and i prove that, but I had an idea ...
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0answers
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How to find the coordinates of a matrix in a new basis?

I have a set of matrices $B=\{A_1, A_2, A_3, A_4\}$ that form a basis in $M_{2\times 2}(\mathbb{R})$. I want to find the coordinates of a matrix $Y_E$ given in the standard basis in this new basis. ...
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1answer
31 views

Find the equation of a plane that is invariant with respect to the following transformation:

Find the equation of a plane that is invariant with respect to the following transformation: \begin{pmatrix}4&-23&17\\ \:11&-43&30\\ \:15&-54&37\end{pmatrix} Actually, I don'...
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0answers
23 views

How is SVD computing unitary square matrices for rank-1 matrices (Matlab)

Consider matrix $\mathbf X=[\mathbf x ~\mathbf x] \in \mathbb R^{D \times 2}$. Of course, $\mathbf X$ has rank-1. Background: $\bullet$The full Singular Value Decomposition (SVD) of $\mathbf X$ is ...
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2answers
45 views

Proof of the existence of a unique linear transformation

I want to prove the following lemma: Let $B$ be a basis for $V$ and let $T_B: B \rightarrow W$ be a map. Then there exists a unique linear map $T_V: V \rightarrow W$ which extends $f,$ that is, such ...
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0answers
24 views

Solving an application with linear transformations

I have sawed this application below on this forum and I wondered If we are looking for a period like "who get the flu after 2 years". We are going to have n=2. And if we don't use eigenvectors. How ...
2
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0answers
23 views

Finding hermitian operator with inner product $\langle p(x),q(x) \rangle = p(0)q(0) + p(1)q(1) + p(-1)q(-1)$

$V=\mathbb{R}_{\leq2} [x]$ with the inner product: $\langle p(x),q(x) \rangle = p(0)q(0) + p(1)q(1) + p(-1)q(-1)$ Let $T: V \rightarrow V$ Linear operator as the following: $T(cx^2 +bx +a) = cx^2 +(...
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1answer
23 views

Geometric effect of linear transformation

Let $T:\Bbb{R}^2\rightarrow\Bbb{R}^2$ be the linear transformation given by $T(x,y)=(x+2y,x).$ Briefly describe the geometric effect of the linear transformation $T$. I found out the matrix ...
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2answers
32 views

Inequality involving ranks [duplicate]

I'm trying to prove the following inequality, $$ \rho(AB) + \rho(BC) \le \rho(B) + \rho(ABC) $$ where $A, B, C \in L(V)$, $V$ is a finite-dimensional vector space and $\rho(A)$ means the rank of ...
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1answer
31 views

Prove that $T$ has at most two distinct eigenvalues [duplicate]

I need help with a problem in Axler's Linear Algebra text. Any hint would be great. Let $V$ be a vector space over $\mathbb{C}$ of dimension $n$, and $T: V\to V.$ If $$\dim \ker(T^{n-2}) \neq \dim \...
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0answers
15 views

Introduction to linear transformation [closed]

I need a introduction to linear transformation. I mean, like a history and usage of the linear transformation.
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1answer
29 views

Different rank and nullity obtained from intuition and computation

I just started learning linear algebra in school. After watching 3Blue1Brown's Essence of Linear Algebra, and there's something I'm confused about. \begin{pmatrix} 2&0\\ -1&1\\ -2&1\\ \...

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