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)
0
votes
0answers
10 views
How to understand root subspace in linear algebra
According to the definition of the root subspace,there exists an exponent k that$(\mathcal A-\lambda\epsilon)^k v=0 $,but how to confirm this exponent?If it is an n*n matrix,what happens when the ...
1
vote
0answers
11 views
Frobenius series solutions and asymptotic
Consider the equation below which is an eigenvalue problem I am studying:
$$
\label{left}
\lambda(v-v'')+c\left(v''-v+be^\xi v+(1-b)e^\xi v'-e^\xi v''\right)'=0.
$$
Restriction on the parameters: $\...
1
vote
1answer
13 views
Interpretation of vectors in dual forms - in matrix equation, and in linear combination of vectors
While a matrix equation $A \vec x=\vec b$ identifies $\vec x$ and $\vec b$ as two vectors, its equivalent form as linear combinations of vectors ${x_1} \vec {a_1} + {x_2} \vec {a_2} = \vec b$ reveals ...
0
votes
0answers
16 views
If there any plane curve whose critical points' curvature are invariant by linear transformation?
I was studying if the curvature of $f$:
$$
f(x) = \frac{ax}{b+x}
$$
can have the critical points located at the same vertical than this other $g$ curve:
$$
g(x) = \frac{ax}{b+x} + cx = f(x) + cx
$$
...
5
votes
1answer
125 views
Matrix representation of Fractional Linear Transformation and the Identity Matrix?
For $x \in \mathbb{R}$, define the fractional linear transformation of $x$ as $f(x)$ where:
$$f(x) = \frac{ax + b}{cx+d}$$
Then $f(x)$ has a matrix representation in $\mathbb{R}^2$ of $F$ where:
...
1
vote
1answer
13 views
Is there an easy way to remove scale from a squared linear transformation matrix
Given a linear transformation matrix $A = \begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{bmatrix}$, I know that one can use SVD or QR decomposition ...
0
votes
0answers
13 views
Coordinate transformation in ODE's with a unit step function
Consider the following general set of 2 ODE's
$$\dot{x}=\Theta(\dot{x} )f_1(x,y)+(1-\Theta(\dot{x}))f_2(x,y)$$
$$\dot{y}=(1-\Theta(\dot{y}))g_1(x,y)+\Theta(\dot{y})g_2(x,y)$$
where $\Theta(x)$ is ...
0
votes
0answers
22 views
If two operators with all eigenvalues real numbers commute then both are triangularizable!
Let $A,B:E\to E$, where $E$ is a finite dimensional vector space, be two linear operators such that all of the eigenvalues are real numbers.
If $AB=BA$, prove that there exists a basis in which ...
0
votes
0answers
14 views
Rotation of vector by rotation matrix
Assume the following expression
$$
\begin{bmatrix}
a_1^* \\
a_2^*
\end{bmatrix}
=
\begin{bmatrix}
\cos(45) & - \sin(45) \\
\sin(45) & \cos(45)
\end{bmatrix}
\begin{bmatrix}
a_1 \\
a_2
\end{...
3
votes
2answers
49 views
How to determine the base of $\ker\phi$ for polynomial function?
Given is a base defined as $$B:=(x\mapsto1,x\mapsto x,x\mapsto x^2,x\mapsto x^3 ,x\mapsto x^4)$$ A set V defined as
$$V:= \{ f: \mathbb{R} \mapsto \mathbb{R}\ |\ \exists\ {a_0},...{a_4} \in \mathbb{R}\...
1
vote
3answers
35 views
Find Bilinear Transformation that maps the points $-i, 0, 2+i$ from $z$ plane on to the points $0, -2i, 4$ of the $ω-$plane?
The solution is supposed to be
$w = \frac{2(z+i)}{z-1}$
Also, is it Okay to ask such questions here? It's not homework, it's the last few sets of problems I can't get on my own. Some belong to ...
0
votes
1answer
12 views
Find Bilinear Transformation that maps the points 0, 1, 2 from z plane on to the points 1, $\frac{1}{2}, \frac{1}{3}$ of the ω-plane [on hold]
Can someone get an answer of ..
$$ω = \frac{1}{z+1}$$
I have tried it 4 times. No luck...
1
vote
3answers
30 views
How to find basis of linear subspace $V$ when $V$ contains all polynomials with the degree up to 4?
I have been given the following definition
$V := \{ f : \mathbb{R} \to \mathbb{R} \mid \textrm{ there are } a_0,\ldots,a_4 \in \mathbb{R}$ and $f(x) = \sum_{i=0}^4 a_i x^i$ for all $x \in \mathbb{R}\}$...
0
votes
2answers
23 views
Find a Linear Transformation with fixed Kernel and Image.
Find a linear transformation $T : \mathbb{R}^4 \rightarrow \mathbb{R}^4$ such that
$$ \operatorname{Ker} (T) = [(1, 0, 0, 1) ; (-1, 0, 0, 1)] $$
$$ \text{Image}(T) = \operatorname{Im}(T) = [(...
0
votes
0answers
15 views
Transformation matrix maps pair of intersecting straight to pair of intersecting straight lines [on hold]
In my Computer graphics mid-term exam, it was asked to prove, Transformation matrix maps pair of intersecting straight lines to intersecting straight lines. Here, transformation matrix is taken in ...
1
vote
1answer
54 views
Why is any rightinverse to T injective? The linear transformation $T$: $\mathbb{R}^5 \longrightarrow \mathbb{R}^4$
Could use some help with this.
The linear transformation $T$: $\mathbb{R}^5 \longrightarrow \mathbb{R}^4$ is given by
$$
T
\left[\begin{matrix}
x_1 \\
x_2 \\
x_3 \\
x_4 \\
x_5
\end{matrix}\...
2
votes
1answer
20 views
Transformation of region
Question:-
Show that the image of the set $ S = {z \in \mathbb{C} | Im(z) \geq 0, |z|\geq 1} $ under the map
$w = u+\iota v = z+ \frac{1}{z}$ is the upper half plane of $v\geq 0$
My approach
Let $ z= ...
5
votes
1answer
83 views
How to define pivot columns?
When you use Gaussian elimination to solve a homogeneous system of linear equations, you end up with "pivot variables" and "non-pivot variables". The non-pivot variables have the property that they ...
0
votes
1answer
35 views
Proving a matrix inequality given another inequality
Suppose that for the $2$-norm, we have $||A||_{2} < 1$. Show that $||I
- A^{T}A||_{2} < 1.$ Assume $A$ is invertible.
I don't know how to solve this problem. I'm studying for an exam. I know ...
0
votes
0answers
25 views
Composition of rotation and reflection
Suppose you find a spatial isometry and you want to classify it. You find out the only real eigenvalue is -1. Also, there is a unique fixed point c.
In this type of cases i was told that we have a ...
0
votes
1answer
26 views
How to get a new matrix with the diagonal elements of another one, and zeros in the rest of the entries? [closed]
I'm NOT interested in how I can do this in MATLAB. I would like to know how, given a matrix $A\in\mathbb{R}^{n\times n}$, I can extract only the elements on its diagonal, put them in a new matrix $B$ (...
0
votes
0answers
32 views
If $\Sigma^{-1}=(A^{-1})^TA^{-1}$, then why does $|A^{-1}|=|\Sigma|^{-1/2}$?
In the derivation of the joint pdf of $f_\textbf{X}(\pmb{x})$, where $\textbf{X}=\pmb\mu+A\pmb Z$ and $\textbf{X}\sim~N_n(\pmb\mu,\Sigma)$, there is a step I do not understand.
In particular, it is ...
1
vote
0answers
37 views
Canonical Jordan form contradiction
I am faced with the following problem:
Given endomorphism $f$ whose characteristic polynomial is $$P_c(x) = (x+1)^{10} (x-1)^{10} x^{10}$$ and whose minimal polynomial is $$P_m (x) = (x+1)^5 (x-1)^...
0
votes
2answers
23 views
Why is the matrix used to describe a linear transformation different if we use row vectors?
Lets say some linear transformation $T$ is defined by: $$T(x, y , z) = (x+2y, 2x+3y-4z, 4x-4y)$$
Now this can be expressed as $\begin{pmatrix}1&2&0\\2&3&-4\\4&-4&0\end{pmatrix}...
0
votes
1answer
41 views
General Gauss-Markov theorem [closed]
$Y=XB+u$ where $X$ is a non random $n\times k$ Matrix, $\textrm{rank}(X)=k, E(u)=0, E(uu')=\sigma^2\Omega$, How to form $(1)$ How to proof $(2)$ the general Gauss-Markov theorem?
0
votes
1answer
32 views
How is inverse of linear operator in $\mathbb{R}[X]$ defined?
Let $T \in \operatorname{End}(\mathbb{R}[X])$
How to think of inverse of $T$ ?
I thought about it in two ways:
$a)\quad \exists U\in \mathbb{R}[X]: Tf\cdot Uf = 1$,
$b)\quad U\left(Tf\right) = f.$
...
0
votes
1answer
48 views
Conjugation of $\left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)$
I'm interested in the following question.
Let $h=\begin{pmatrix} 0 & -1 \\ 1 & 0 \\ \end{pmatrix}$. This is an orthogonal map which is quite far away from the identity (say in the Frobenius ...
0
votes
1answer
35 views
Difference between two Matrix representation of Linear Transformation
I have a linear transformation $$T(x_1,x_2,x_3) = (x_1-x_2+2x_3,2x_1+x_2,-x_1-2x_2+2x_3) $$
Assuming an ordered basis $\{u_1,u_2,u_3\}$ I get the linear transformation of the basis vectors as below: $...
0
votes
2answers
18 views
Question on bilineal and quadratic forms
My first question is:
Let $\phi: V \rightarrow \mathbb{R}$ a quadratic form. We say that a vector $x$ is autoconjugate if its conjugate with its self, that is, $\phi(x)=0$. Does the set of all ...
-1
votes
0answers
9 views
How to Create a Homography Matrix in Matlab from n Pairs of Points?
I'm trying to create a homography matrix using the procedure shown below within Matlab:
However, I'm having trouble actually coding it. I don't want to use SVD if possible and I'm not allowed to use ...
1
vote
1answer
29 views
Product of two Linear Operators
Let $S(e_n)=e_{n+1}$ and $T(e_n)=e_{n+2}$ be two linear operators on the Hilbert space $l_2(N)$, the space of all sequences $\sum_{1}^\infty |a_k|^2 < \infty$, and $\{e_n\}, n=0,1,2,...$ is the ...
0
votes
2answers
67 views
Proving a projection onto subspace if $P^2 = P$
Suppose that V is a vector space, and M is a subspace of V . A transformation $P : V \to V$
is called the projection of V onto M if
(i) there exists a subspace N such that every vector v ∈ V can be ...
0
votes
1answer
21 views
General Formula of a Linear Operator given its act on the Standard Orthonormal Basis
I am trying to find a general formula for a linear operator on a Hilbert space when its action on the standard orthonormal basis is known.
I include my work below. Please tell me whether my solution ...
0
votes
1answer
74 views
Eigendecomposition of a matrix (what assumptions need to be made)
For the sake of brevity, I'm given a 3x3 symmetric matrix with real entries with no further information as to what the rows and columns encode. (eg. this not need to be the case but the columns may ...
0
votes
0answers
20 views
Orthogonal Transformation Matrices issue
I'm really confused. If $V$ is an orthogonal matrix that transforms matrix $U_1$ into matrix $U_2$ why is it that $U_2 = V^TU_1V$?
With what I know, an orthogonal matrix is one such that $V^{-1} = V^...
1
vote
0answers
7 views
Looking for ways to transform time-series data recorded from object movement into equation describing the movement direction of the object
Looking for some time-series data transformation advice!
I want to know what's the best way to transform data of 9-tuples time series data of IMU (Inertia Measurement Unit) sensor, recorded from a ...
0
votes
1answer
38 views
Showing isomorphism in transformation between matrix and bilinear map spaces
From Serge Lang Linear Algebra:
Show that the association $A \rightarrow g_A$ is an isomorphism
between the space of $m \times n$ matrices, and the space of bilinear maps of $\mathbb{K}^m \times ...
1
vote
1answer
18 views
Find a basis for witch the matrix consists only of 0 and 1
Let V and W be any two finite dimensional vector spaces over a field F.
Let T : V →W be a linear map. Prove that there are bases of V and W
such that with respect to those bases T is represented by ...
0
votes
0answers
19 views
Encryption and Decryption of Alpahabets to Integers
I'll explain my case with an example. Say, I've 2 words,
FOOTBALL
FOOD
Let f(FOOTBALL) = 845614,
f(FOOD) = 74312
and f(FO) = 132
Is there any method to get
f'(845614, 2) = 132 and f'(...
1
vote
1answer
24 views
(When) is the bicommutant of a linear operator equal to the set of “simpler” operators?
Let $V$ be a $\Bbbk$-linear space and $T\in \mathrm{End}_\Bbbk(V)$.
Definition. Say an operator $S\in \mathrm{End}_\Bbbk(V)$ is simpler than $T$ if every $T$-invariant (internal) direct sum ...
5
votes
1answer
59 views
Average area of the shadow of a convex shape [closed]
What is the average area of the shadow of a convex shape taken over all possible orientations?
If we take a sphere, its surface area is exactly 4 times the area of its shadow. How can it be ...
1
vote
1answer
23 views
Sequences of Continuous Linear Operators between Banach Spaces.
Let $E, F$ be two Banach Spaces. Let $\{ T_{n} \}$ be a sequence of continuous linear operators from $E$ into $F$ such that:
For all $x \in E: T_{n}x \rightarrow Tx,$ some limit in $F$.
Then the ...
0
votes
2answers
236 views
Linear Algebra - Proving a projection given a linear transformation
Suppose that $V$ is a vector space, and $M$ is a subspace of $V$.
A transformation $P:V \rightarrow V$ is called the projection of $V$ onto $M$ if
(i) there exists a subspace $N$ such that every ...
0
votes
2answers
75 views
Linear Algebra - Proving a projection onto a subspace is a linear transformation
How do I prove that a projection onto a subspace is a linear transformation ?
Given that V is a vector space, and M is a subspace of V.
I know these two facts:
i) There exists a subspace $N$ such ...
0
votes
1answer
26 views
Understanding the association of the bilinear form and matrix associated with it
From S.L Linear Algebra:
Show that the association $A \rightarrow g_A$ is an isomorphism
between the space of $m \times n$ matrices, and the space of bilinear maps of $\mathbb{K}^m \times \...
0
votes
0answers
27 views
Monotonic linear interpolation of monotonic data on a scattered grid
Let $x_1,\dots,x_n\in[0,1]^d$ and $y_1,\dots,y_n\in[0,1]$ satisfy:
$$\forall i,j\in {1,\dots,n},\hspace{10pt} x_i \le x_j \rightarrow y_i \le y_j.$$
I seek to find a continuous function $f:[0,1]^d\...
0
votes
2answers
99 views
Linear Algebra - Invariant Subspaces in 2-dimensions
How do I find 2-dimensional subspaces that are invariant subspace of T ?
$T\left(\left[
\begin{matrix}
x\\
y\\
z\\
\end{matrix}
\right]\right)
=
\left[\begin{matrix}
2y+z \\
-2x+4y+z \\
-2x+2y+3z
\...
1
vote
1answer
28 views
Meaning of the determinant of the restriction of a linear map
Suppose $T : \mathbb{R}^n \to \mathbb{R}^n$ is a linear map and let $U \subset \mathbb{R}^n$ be a $d$-dimensional subspace where $0 < d < n$ and $\ker T = U$. I was wondering how to make sense ...
3
votes
0answers
45 views
What's the geometric interpretation of the square root of a matrix?
Question:
If I have a matrix $A$, I know that its square root is a matrix that has the same eigenvectors as $A$ but its eigenvalues are the square roots of the eigenvalues of $A$.
What does this ...
0
votes
0answers
20 views
Doubt on range-space of linear transformation
I have a linear transformation $T:V\rightarrow W$. I believe that if T is onto, then, range space of T is equal to W. My reasoning is that if T is onto, then every member of W has a pre-image. So, W ...
