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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Set of all endomorphisms that commute with given endomorhphism $F$, with $\mu_F = \chi_F$ is the same as $\mathbb{k}[F]$.

Let $F$ be an endomorphism of $V$ over some field $\mathbb{k}$ and characteristic polynomial of $F$ is the same as its minimal polynomial. Is that true that $Z_F \overset{\operatorname{def}}{=} \{G \...
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How can I rotate a vector in a spiral motion around the z-axis?

I posted a similar question on the programming board, but I thought it might be more fitting here; I'm trying to gradually rotate a vector back to some initial angle w.r.t the Z axis after having ...
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a question about the gemoetric represntation of non-squre determinant [closed]

sadlly in class all the professor showed me was theortical so im trying to learn the geometry of all the actions alone. i understand that the determinant is a way to explain the size of a 1 by n times ...
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Finding the matrix of a linear transformation in a basis

Let $C$ be a basis for $\mathbb{R}^3$. $$C = \{c_1 = \begin{bmatrix} 1 \cr 2 \cr 3 \end{bmatrix}, c_2 = \begin{bmatrix} 0 \cr 1 \cr -1 \end{bmatrix}, c_3 = \begin{bmatrix} 0 \cr 0 \cr 1 \end{bmatrix}\}...
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What is the connection between extrinsic and intrinsic rotation?

The image below shows the frames $F1$ and $F2$ as well as the rotation matrix $R1$ which is a 90 degree rotation about the z axis of $F1$: $$R1 = {}^{F_1}R_{F_2}(\pi/2)_z$$ Now let's say I wanted to ...
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A proof that the null linear mapping is the only one whose matrix representation does not depend on the basis [FALSE]

I would like to show the fact that the linear mapping $$ L : E\to E $$ $$ x\mapsto0_{E} $$ is the unique linear mapping whose matrix representation does not depend on the choice of the basis. My ...
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Notation: set of marices with left/right inverse

We usually denote $GL_n(\mathbb{K})$ to the set of all invertible matrices of dimension $n \times n$ over field $\mathbb{K}$. But is there a notation for all $n\times n$ matrices over $\mathbb{K}$ ...
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How do I find the associate matrix of $f(A) = A \begin{bmatrix} 1 & 2 \\ 3 & \lambda \end{bmatrix}$

I need to find the associate matrix of this linear function ($A$ is a 2x2 matrix) $f:M_{2\times 2}(\mathbb{R})\to M_{2\times 2}(\mathbb{R})$ $f(A) = A \begin{bmatrix} 1 & 2 \\ 3 & \lambda \...
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How can I find the unit vector from a rotation matrix? [closed]

I am confused about how to find the unit vector from a rotation matrix. If $R$ is a $3\times 3$ rotation matrix, $$R = \begin{bmatrix} 0.5 & 2 & 1 \\ 2 & 1 & -0.5 \\ -0.5 & 1.5 &...
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Is there a unique solution? inv(T)*A*T= B [closed]

If A, B, and T are all invertible rigid-body transformation ($A, B, T \in SE(3)$, all represented as $4 \times 4$ matrices with bottom row = [0 0 0 1]) and I know A and B and want to know T, does ...
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Matrix representation of linear map $T: \mathbb{C}^2 \to \mathbb{C}^2$. Where's the error?

Let $B=\{(1, i), (-i, 2)\}$ be a basis of $\mathbb{C}^2$ and $T:\mathbb{C}^2 \to\mathbb{C}^2$ the linear map $$T(x, y) = (x, 0)$$ Find a matrix representation of $T$ with respect to basis $B$. ...
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How can I find the matrix of affine transformations?

I know the points of a triangle: x1 = (0,2), x2 = (-2,-2) and x3 = (2,-2) and the triangle has been scaled, translated and rotated, how can I find the scaling, translation and rotation matrices (...
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Inverse of a linear map involving polynomial derivatives

Let $V = \bigl\{f:\mathbb{R} \to \mathbb{R} \mid \exists a_0,\dots,a_4\in \mathbb{R},\, f(x)=\sum_{i=0}^{4}a_ix^i \bigr\}$, we define the linear map $\phi:V \rightarrow V$ as follows: $$ \phi(f)(x)=f''...
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Can row space be understood as a set of linear combination that represents a single solution?

I doubt that I understand row space. I'm not sure what it means to have a solution in the row space of matrix $A$. This seems to mean that the linear combination of the rows forms a solution of $Ax=b$....
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Finding the matrix for the reflection of a vector over some arbitrary line in $\mathbb{R}^2$

With a little bit of drawing, I derived the matrix for the projection of some arbitrary vector $\vec{x} \in \mathbb{R}^2$ onto some line $L$, where $\vec{u} = \begin{bmatrix} u_1 \\ u_2\end{bmatrix}$ ...
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An example of transformation matrix of polynomial derivatives.

Let $V=\{f:\mathbb{R}\rightarrow \mathbb{R}| \exists a_0,...,a_4\in \mathbb{R},f(x)=\sum_{i=0}^{4}a_ix^i \}$, we define the linear map $\phi:V \rightarrow V$ as follows: $$\phi(f)(x)=f''(x)+xf'(x)-f(x+...
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Why does the sum of eigenvalues equal to trace in terms of linear transformations?

While studying eigenvectors, I was confronted with two statements: The product of the eigenvalues of some matrix $A$ is equal to the determinant of $A$ The trace of $A$ is equal to the sum of its ...
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Proving that $rank(A) + rank(B) - n \le rank(AB)$ [duplicate]

I have to demonstrate the following statement: If $\ A \in M_{m,n}(\mathbb{R})$ and $\ B \in M_{n,t}(\mathbb{R})$ show that: $$ rank(A)+ rank(B) -n \ \le \ rank(AB) \ \le \ \min\{rank(A), rank(B)\} $...
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Papa Rudin $4.12 $ theorem

There are things that we need for the proof of the theorem: ]1 There is the theorem: If $L$ is a continuous linear functional on $H$, then there is a unique $y$ $\in$ $H$ such that $Lx$ $=$ $(x,y)$ ($...
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Proving there exists basis $B$ for $R^2$ such that $[T]_B = \begin{bmatrix}0 & 0 \\ 0 & 2 \end{bmatrix}$

I've been stumped by this question for the last 2 days, and couldn't figure how to prove it. $T:R^2 \to R^2$ is a linear transformation (T$\neq$0) that satisfies $T^2=2T$ and it is known that $T$ is ...
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Gaussian Elimination to find the intersection point between two planes [closed]

I want to ask if it is possible to find the intersection point between two planes using the Gaussian Elimination method.
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Transformation matrix with respect to two bases

Let $V=\mathbb{Q^{2\times3}}$ , $W=\mathbb{Q^{2\times2}}$be vector spaces over $\mathbb{Q}$ and consider the linear map $\phi:V\rightarrow W$ given by: $$\phi(M)=MA, \ A=\begin {pmatrix} 1&-3 \\2 &...
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When a faithful linear Lie algebra representation preserves composition?

Let $L$ be $\mathfrak{sl}(2,F)$ with the basis $$ x=\pmatrix{0&1\\0&0},~y=\pmatrix{0&0\\1&0},~h=\pmatrix{1&0\\0&-1}. $$ From Humphreys, on classification of irreducible $L$-...
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If A is a positive linear transformation, AB is self-adjoint, then $|(ABx,x)| \leq ||B||(Ax,x)$ or $|(ABx,x)| \leq \rho(B)(Ax,x)$

Prove or disprove: If $A$ is a positive linear transformation, $AB$ is self-adjoint, then a, $|(ABx,x)| \leq ||B||.(Ax,x)$ b, $|(ABx,x)| \leq \rho(B).(Ax,x)$ With the matrix norm defined by: $||A|| := ...
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Building topological equivalence between two vector fields

I'm reading about topological equivalence between different vector fields and I would like to know how to build these mappings. Let us consider one pedagogical example, suppose I have the following ...
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How to get the state transition matrix by fundamental matrix when the fundamental matrix is not invertible? [closed]

I want to get the state transition matrix by the fundamental matrix solution $U(x)\in\mathbb{R}^{2\times2}$ as follow $\Phi(x,x_0)=U(x)U^{-1}(x_0)$. But since one column of $U(x_0)$ is $0$, the ...
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How to use matrices to rotate two matrix to specific positions while keeping their relative position unchanged?

Suppose there are two vectors $\overrightarrow{a}, \overrightarrow{b}$ that define a plane. How do I find matrices that can be applied to both verctors, so that $\overrightarrow{a}$ will be rotated to ...
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Is being isomorphic to dual hereditary?

Let $V$ be a vector space, and assume that $V$ is isomorphic to its dual, i.e., $V \simeq V^*$. Is every linear subspace $U$ of $V$ also isomorphic to its dual, i.e., $U \simeq U^*$? This is certainly ...
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From Vandermonde matrix to Newton basis polynomial matrix (lower triangular)

Suppose I have an $n\times k$ Vandermonde matrix $V$ where the roots/points in $x \in \mathbb{R}^n$ are all distinct and matrix $V$ is of the form $$V= [1 \; x \; x^2 \ldots x^{k-1}]$$ where the ...
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Does the direct sum of the image and kernel of the linear transformation form the original space? [duplicate]

Does the direct sum of the image and kernel of the linear transformation equal to the original space? I know if the linear transformation is a projection, the direct sum of the image and kernel of the ...
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Action of Linear Operator on basis of a vector space.

Say I have a vector space $V$ having basis vectors $\vec{e}_i \text{ for } 1\le i\le n$. Then, the action of a linear operator $f$ on the basis vectors is given by: $$f(\vec{e}_i) = \sum_{k=1}^n f_{ki}...
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Is there any simple set of properties that uniquely characterizes differentiation in the space of complex functions?

The transformation of differentiation is a linear operator over the vector space of entire functions (call this space $\mathbb{C}^E.$) Is there any simple set of properties that uniquely determines ...
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Finding a single matrix-element of a linear transformation with a change of basis

The question: Let $A = \begin{pmatrix} a & b & c\\ d & e & f \\ g & h & i\end{pmatrix}$ be the standard matrix of linear transformation $T : \mathbb{R}^3 \rightarrow \mathbb{R}^...
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Papa Rudin Theorem $4.11$ $(c)$ and $(d)$.

This is the definition which we need for the proof: There is the theorem: Let $M$ be a closed subspace of a Hilbert space $H$. $(a)$ every $x$ $\in$ $H$ has then a unique decomposition $x$ $=$ $Px$ $+...
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Visualizing Linear Transformation of Unit Circle via Matrix Multiplication and Eigenvectors

Consider the matrix \begin{equation*} A = \begin{pmatrix} 5 & 1\\ 1 & 5\\ \end{pmatrix} \end{equation*} $\textbf{Question 1:}$ Draw the image of a unit circle after multiplying with ...
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Proving linear transformation on $C[0,1]$ is not invertible

Let $X=C[0,1]$ be the space of continuous functions on the interval $[0,1]$, $t_0\in J$. I am trying to show that if $T: X\to X$ defined by $Tx=vx$, where $v\in X$ is fixed, and $v(t_0)=0$, then $T$ ...
2 votes
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Normal Form of Linear Maps Between Matrices

I am looking for a reference which features the following result -- for lack a better term I will call this "normal form" (of a linear map between matrices) -- and which explores its ...
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Linear Algebra-Linear Operator

How do i construct a linear operator of rank=2 from R^3 to itself. Is it sufficient if i find a linear operator and deduce that it is of rank 2 and it fulfills the conditions of being a linear ...
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How to find the coordinate transformation from 1 metric to anither [closed]

Let’s say I have a metric tensor in a coordinate system $(t,x,y,z)$ and the same metric tensor expressed in another coordinate system $(t^\prime,a,b,c)$. How do I find the corrdinate transformation (...
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Name of such column-permuting Matrix

Let $\vec{v_1}$ be an $n$-vector with nonnegative entries. Let $\vec{v_i}$, where $i=2,3,\cdots, n$, be an $n$-vector generated from $P_iv_1$ where $P_i$ is a permutation matrix such that $P_i\neq P_j$...
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Verification of Accelerometer offset calculations occurring due to placement of Accelerometer away from center of rotation in a rigid body

I have evaluated the Accelerometer offset occurring due to placement of Accelerometer away from the centre of rotation of body. In the below evaluation I am trying to calculate Accelerometer's reading ...
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How to optimize parameters with orthogonal constraints?

I'm working on a Compute Vision problem. It can be defined by the formula below: $$ minimize \ f(T) \\ s.t \ T = \begin{bmatrix} R & t \\ 0 & 1 \\ \end{bmatrix}, R^TR = I, {det}(R) = 1 ...
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Expressing a $2\times 2$ matrix as a single equation.

From another question (Verifying a Linear transformation from M 2,2 -> R) I saw the following proof which shows that $$T(kv)=kT(v)$$ $$T(kv)=T\left(\begin{bmatrix}k\cdot a&k\cdot b\\k\cdot c&...
3 votes
2 answers
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Linear transformation and direct sum

When given $T:V \to V$ such that $T^2=I$, prove that $U,W \subseteq V$ subspaces exist such that: $$V=U\oplus W$$ $$T(u)=u, \forall u \in U$$ $$T(w)=-w, \forall w \in W$$ My first thought was to ...
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1 vote
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Linear maps from finite-dimensional spaces are continuous. Proof?

I am trying to find a non-circular path that leads me to the following result: Theorem 1. Linear maps from a finite-dimensional normed linear space (to possibly infinite dimensional spaces) are ...
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Properties of a reflection matrix

If you are given a transformation matrix, how do you check if it represents a reflection quickly? Or if it represents rotation? What is uniq about them that I can easily spot(it doesn't need to be ...
1 vote
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How to show that the column spaces of two matrices is equal?

The linear transformations $R^3 \rightarrow R^3$ have the matrices $A= \begin{bmatrix} 1&2&0 \\ 2&3&3\\ 1&a&3 \end {bmatrix}$ and $B = \begin{bmatrix} 4&1&3 \\ 7&1&...
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Which matrices preserve range?

Any matrix $M$ has a given range (column space). Let $B$ be a matrix such that for all $M$, $MB$ has the same column space as $M$. Clearly, the identity matrix $I$ satisfies this, as does any non-...
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2 answers
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Why do eigenspaces attract a sequence?

I was reading about eigen-stuff and I came across a very interesting visualization. We know that, a matrix when multiplied with a vector in a $2D$ space simply maps that vector to someplace else, ...
1 vote
1 answer
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How to find the transformation matrix for rotation?

I have a basis B in $R^n$ made of orthogonal vectors with the norm 1 (ON-Basis). Lets call them $v_1$, $v_2$ $v_3$. Find the transformation matrix for when $v_1$ rotates counterclockwise 45 degrees. $...

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