Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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)

29
votes
5answers
71k views

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 ...
4
votes
2answers
7k views

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 ...
3
votes
1answer
1k 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{...
4
votes
3answers
445 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 ...
2
votes
0answers
207 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 ...
17
votes
3answers
126k views

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{...
4
votes
2answers
1k views

Prove that $v, Tv, T^2v, … , T^{m-1}v$ is linearly independent

Suppose $T$ is in $L(V)$, $m$ is a positive integer, and $v$ in vector space $V$ is such that $(T^{m-1})v \neq 0$, and $(T^m)v = 0$. Prove that $[v, Tv, T^2v, ... , T^{m-1}v]$ ...
39
votes
6answers
10k 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 + ...
7
votes
2answers
1k views

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 ...
2
votes
3answers
452 views

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\...
2
votes
1answer
263 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\{...
2
votes
1answer
928 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 ...
0
votes
2answers
2k 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 ...
3
votes
2answers
218 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 (...
0
votes
1answer
80 views

Inverse of Linear Transformations

For each of the following linear transformations, find the inverse if it exists, or explain why there is no inverse. (a) T : R 3 → R 3 where T(v) is the reflection of v around the plane x + 2y + 3z = ...
0
votes
1answer
206 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 ...
18
votes
2answers
3k 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 \...
17
votes
7answers
13k 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 ...
9
votes
4answers
773 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 ...
4
votes
1answer
892 views

Proof general state space similarity transformation to controllable canonical form

Given a state space model of the form, $$ \begin{align} \dot{x} &= A\,x + B\,u \\ y &= C\,x + D\,u \end{align} \tag{1} $$ however I think that this would also apply to a discrete time model. ...
12
votes
5answers
217 views

Ill-known/original/interesting investigations on/applications of inversion (the geometric transform)

Inversion transform with center (or pole) $C$ and power $k^2$ is defined by: $$\tag{1}J_{C,k}:M \leftrightarrow M' \ \ \ \ \ \iff \ \ \ \ \ \ \ \vec{CM'}=\frac{k^2}{||\vec{CM}||^2} \ \vec{CM} $$ It ...
4
votes
5answers
4k views

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 \...
1
vote
1answer
340 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 ...
5
votes
0answers
164 views

Linear transformation $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $T^{2}=\lambda T.$ [closed]

Let $T$ be a linear transformation $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $T^{2}=\lambda T$ for some $\lambda\in\mathbb{R}.$ Then which of the following is/are true? $1.\|T(x)\|=|\...
4
votes
3answers
584 views

Show that $\ker \hat{T} = \text{ann}(\text{range } T)$

This is an old exam problem: Let $V$ and $W$ be finite dimensional vector spaces over a field $F$ and let $T: V \to W$ be a linear transformation. Define $\hat{T}: W^* \to V^*$ by $(\hat{T}(f))(v)=f(T(...
4
votes
1answer
107 views

If $k$ is an eigenvalue of $A$ of algebraic multiplicity $r$, then is $p(k)$ an eigenvalue of $p(A)$ of algebraic multiplicity $r$?

Let $k \in \mathbb C$ be an eigenvalue of $A \in M(n,\mathbb C)$ of algebraic multiplicity $r$ (i.e. $k$ is an $r$-fold root of the characteristic polynomial of $A$). Let $p(x)$ be a polynomial with ...
4
votes
2answers
225 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, ...
4
votes
0answers
2k views

Reducing or avoiding the Gibbs phenomenon.

What is your favourite method which would help reduce the Gibbs phenomenon in Fourier Series and Fourier Transforms. This could mean pre-processing or post-processing or altering the transform. With ...
3
votes
2answers
153 views

Prove that T is a linear transformation

Does it matter that in the first line it's written $T(\alpha p+ \beta g)$ and not $T(\alpha p(t)+ \beta g(t))$ but at the end it is written with $\alpha T(p(t)) + \beta T(g(t)))$ with the $t$'s. ...
2
votes
1answer
439 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 ...
1
vote
1answer
96 views

Show a subset is subspace of a vectorspace and find the dimension

Let $n>1$ be a natural number and let $\alpha\in\mathbb{R}$ be a real scalar. Let $V$ be a subset of the vector space $P_n(\mathbb{R})$. Define $V$ as $$V=\{p\in P_n(\mathbb{R}):p(\alpha)=0\}$$ ...
1
vote
5answers
802 views

Two idempotent matrices are similar iff they have the same rank

Definitions: For $F$ a field, $A,B\in F^{m\times n}$ are equivalent means there exist invertible matrices $Q\in F^{m\times m}$ and $P\in F^{n\times n}$ so that $B=QAP$, $A,B\in F^{n\times n}$ are ...
1
vote
3answers
2k 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 ...
0
votes
1answer
306 views

Find matrix of linear transformation relative to new bases

If $T:\Bbb R^3\to \Bbb R^2$ is a linear transformation, and the matrix of $T$ = $\left(\begin{array}{ccc}0 & 1 & 1 \\0 & 1 & -1\end{array}\right)$. If you use the basis $\{i,j,k\}$ ...
0
votes
1answer
514 views

Linear functional f(v)=0 imply v=0

Good day, I want to prove the following theorem: 1) For any nonzero vector $ v \in V $, there exists a linear funtional $ f \in V^*$ for wich $f(v) \neq 0 $ I know that if $f$ is a lineal functional ...
0
votes
1answer
4k views

Reflect point across line with matrix

What is the transformation matrix that I multiply a point by if I want to reflect that point across a line that goes through the origin in terms of the angle between the line and the x-axis? In other ...
-1
votes
4answers
267 views

How to find the matrix representation of a linear tranformation

How does one find the matrix representation of a linear transformation $T:V\to W$ with respect to the basis $B$ for $V$ and $D$ for $W$?
48
votes
9answers
3k views

If the field of a vector space weren't characteristic zero, then what would change in the theory?

In the book of Linear Algebra by Werner Greub, whenever we choose a field for our vector spaces, we always choose an arbitrary field $F$ of characteristic zero, but to understand the importance of the ...
34
votes
6answers
3k views

Prove that if V is finite dimensional then V is even dimensional?

Let $f:V \to V$ be a linear map such that $(f\circ f)(v) = -v$. Prove that if $V$ is a finite dimensional vector space over $\mathbb R$, $V$ is even dimensional. From what I can figure out for myself,...
28
votes
3answers
2k views

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 ...
13
votes
2answers
636 views

Intuitive geometric explanation: existence of eigenvalue in odd dimension real vector space.

I'm looking for an intuitive geometric explanation for the fact that given an odd dimensional real vector space $W$ and an endomorphism $T:W \rightarrow W$, there exists a real eigenvalue of $T$. I'm ...
6
votes
2answers
208 views

What is the mechanism of Eigenvector? [closed]

I have studied EigenValues and EigenVectors but still what I can't see is that how EigenVectors become transformed or rotated vectors.
11
votes
2answers
2k views

Intuition behind the definition of linear transformation

I have studied that given vector spaces $V_1$ and $V_2$, a function $T:V_1 \rightarrow V_2$ is called a linear transformation of $V_1$ into $V_2$, if following two properties are true for all $u, v \...
7
votes
2answers
3k 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 ...
5
votes
3answers
388 views

Is self-adjointness really a property of an operator, or of an operator and an inner product?

On a Hilbert- (or otherwise inner-product-) space $\mathcal{H}$ with scalar product $S = \langle.|.\rangle_\mathcal{H}$, a self-adjoint operator is is readily defined as a linear mapping $A : \mathcal{...
3
votes
1answer
2k views

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 ...
3
votes
3answers
361 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 ...
6
votes
2answers
2k views

examples of linear map $f:V \rightarrow V$, which is injective but not surjective

I am trying to find a linear map $f:V \rightarrow V$, which is injective but not surjective. I always thought that if the dimension of the domain and codomain are equal and the map is injective it ...
5
votes
1answer
809 views

Exact same solutions implies same row-reduced echelon form?

In Hoffman and Kunze they have two exercises where they ask to show that if two homogeneous linear systems have the exact same solutions then they have the same row-reduced echelon form. They first ...
5
votes
1answer
430 views

Dimension of $\mathbb{Q}$-vector space if a nonsingular linear transformation $T$ exists such that $T^{-1} = T^{2} + T$

We're given $V$ a finite dimensional vector space over $\mathbb{Q}$, $T$ a non-singular linear transformation of $V$ such that $T^{-1} = T^{2} + T$. The question has two parts. If I understand part (a)...