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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Question about symmetric linear transformation
In the real inner product space $V$, a linear transformation $A$ is a symmetric transformation if $\langle A\alpha,\beta\rangle =\langle\alpha,A\beta\rangle$
In my book it says a linear ...
3
votes
2answers
29 views
Finding the characteristic polynomial of the $T: \mathcal{M}_{n} (\mathbb{R}) \to \mathcal{M}_{n} (\mathbb{R})$ given by $T(M)=M^{\text{tr}}$
Here's a problem from Larry Smith's Linear Algebra textbook:
Let $\mathcal{M}_{n} (\mathbb{R})$ be the set of real matrices of
order $n \times n$. Let $T: \mathcal{M}_{n} (\mathbb{R}) \to \...
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1answer
25 views
Find kernel and image of linear tranformation from matrix
Given following linear transformation in matrix space 2x2:
$$T(X) = \begin{pmatrix} 1 & 7 \\ 7 & 2018 \end{pmatrix}X - X\begin{pmatrix} 1 & 7\\ 7 & 2018 \end{pmatrix}$$
How to find ...
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0answers
24 views
linear transformation between two bases for the polynomial space $P_2[x]$
$B=\{1,1+x,1+x+x^2\}, C=\{1+x,2x,x^2-1\}$
$B$ and $A$ are both bases for the polynomial space $P_2[x]$.
$T: P_2[x] \to P_2[x]$
$$[T]B =
\begin{pmatrix}
3 & 3 & 3\\
-2 & -...
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0answers
10 views
Representing a transformation's matrix as an inner product
Let $V$ be an inner product space, $T: V \to V$ a linear map, and $A=M(T,P,P)$ for an orthonormal basis $P=[v_1,...,v_n]$ (where $M(T,P,Q)$ is the matrix representation of $T$ with respect to $P$ and $...
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votes
1answer
31 views
Matrices linear algebra linear transformations [on hold]
Prove that following statements are equivalent for A€M(C).
1. A is invertible
2. Coloumns of A are linearly independent.
3. The rows of A are linearly independent.
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1answer
25 views
For A,B subsets of a Normed Vector Space, A closed, B Compact, Show A - B Is Closed [duplicate]
Statement of the problem:
Let $E$ be a Normed Vector Space over the real numbers. Let $A, B$ be subsets of $E$ such that:
$A$ and $B$ are non-empty, $A \cap B = \emptyset $. Assume $A$ is closed and ...
2
votes
1answer
42 views
Deeper understanding of the adjoint of a linear operator
My undergraduate classes in Q.M describes the adjoint of a linear operator purely as a mathematical formality.
At this point, I'd like a deeper and heuristic understanding of it.
My questions are ...
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2answers
24 views
For some linear transformation T, if $T(u_0)=w$ and $T(v)=0$ then $T(u)=w$ if and only if $u=u_0+v$ proof inquiry
Let $T : V → W$ be a linear transformation. Let $w ∈ W$ and let $u_0 ∈ V$ satisfy
$$T(u_0) = w$$
Show that $u ∈ V$ is a solution of the equation
$T(u) = w$ if and only if $u = u_0 + v$, where $T(v) =...
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4answers
35 views
Uniqueness of a linear map on a basis of a vector space
From Linear Algebra Done Right, 3rd edition, by Sheldon Axler:
Suppose $v_1, \ldots, v_n$ is a basis of $V$, and $w_1, \ldots, w_n \in W$. Then there exists a unique linear map $T: V \to W$ such ...
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1answer
34 views
Find charateristic polynomial for linear operator
I have a linear operator $L:P_2(\mathbb{R})\to P_2(\mathbb{R})$ where
$L(\alpha+\beta X)=(3\alpha+2\beta)+(\alpha+2\beta)X, \alpha ,\beta\in\mathbb{R}$,
I want to find the following for $L$:
The ...
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0answers
20 views
Group theory : representation Matrix of generating element for families of functions?
Assume I have some sampling of a function $f(t)$ at points $t$:
$$f(t_k) = d_k, \forall k \in \{1,\cdots,n\}$$
Assume we have vectors $\bf v_k$ which can be functions of $x_k$ and $d_k$, can we find ...
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0answers
19 views
There is no T-invariant subspace $U$ such that $\mathbb{R}^{3} = W\oplus U$
Could someone give me a suggestion to solve the following problem problem?
PROBLEM. Let $T : \mathbb{R}^{3} \longrightarrow \mathbb{R}^{3}$ and $\beta$ a basis of $\mathbb{R}^{3}$ such that
$$ \left[ ...
1
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1answer
43 views
Dimension of kernel subspace of trace transformation
From S.L Linear Algebra:
Let $V$ be the vector space of real $n \times n$ symmetric matrices.
What is $\textrm{dim} \, V$? What is the dimension of the subspace $W$ consisting
of those matrices ...
2
votes
3answers
77 views
What two transformations are described by $A=\begin{bmatrix} 2 & 4 \\\ 1 & 2 \end{bmatrix}$
Apparently the following matrix
$$A=\begin{bmatrix}
2 & 4
\\\ 1 & 2
\end{bmatrix}$$
is a composition of two linear transformations, $BC=A$.
The objective of this exercise is to decompose $A$...
0
votes
1answer
25 views
Show that a linear map is one-to-one if and only if it preserves linear independence.
From Jim Heffron's free textbook Linear Algebra: 3rd Edition, question 3.2.27 states: "Show that a linear map is one-to-one if and only if it preserves linear independence."
The answer key, also ...
2
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3answers
48 views
Will the product of a real valued matrix and its transpose always have a real eigenvector?
I'm trying to solve this really interesting problem, taken from "Berkeley Problmes in Mathematics" by Souza and Silva
https://www.amazon.com/Berkeley-Problems-Mathematics-Problem-Books/dp/0387204296
...
1
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1answer
40 views
Help a robot avoid computing the matrix inverse using the Gram-Schmidt process
Imagine you have a robot whose position is recorded as $t_1, t_2, t_3, \dots$ in its coordinate frame. Check the visualization here.
We can write down the robot's basis in our 2D image plane, i.e. ...
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0answers
18 views
How to find all the derivation of matrix algebra
Suppose F is a field, and $M_n(F)$ is a matrix algebra over F, denoted by V. If $\phi \in L(V)$, the set of all linear map between V, and $\phi$ satisfies Leibeniz's law: $\phi(AB) = \phi(A)B+A\phi(B)$...
1
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1answer
18 views
all 2 dimensional invariant subspaces
How we can find all 2 dimensional invariant subspaces of
\begin{pmatrix}
2 & 1 & 0 \\
0 & 2 & 0 \\
0 & 0 & 8
\end{pmatrix}
I know that there are at least 2 such subspaces, ...
4
votes
4answers
75 views
Is it true that the eigenvalues of $A + B$ are the sum of some eigenvalue of $A$ and some eigenvalue of $B$?
Is it true that the eigenvalues of $A + B$ are the sum of some eigenvalue of $A$ and some eigenvalue of $B$?
I'm taking a linear algebra class, and I recently learned about eigenvalues. I think that ...
0
votes
2answers
44 views
Computing the matrix exponential for a Jordan matrix
How can I compute $e^{At}$ where $A = J_{3}(5)$? That is,
$$A = \begin{pmatrix} 5 & 1 & 0 \\ 0 & 5 & 1 \\ 0 & 0 & 5 \end{pmatrix} $$
Using this, how can I write down a basis ...
2
votes
0answers
22 views
Basis for $\mathcal{L}(E)$ formed by projections
Let $E$ be a finite dimensional vector space. Find a basis of $\mathcal{L}(E)$ formed by projections. a.k.a : endomorphisms p such that $p^2=p$.
My solution was to let $(e_1,...,e_n)$ be a basis for ...
0
votes
5answers
55 views
If $AB=I$ show that $A$ has full rank.
Let $A,B\in \mathbb R^{n\times n}$. Show that if $AB=I$ then $B$ has full rank.
In fact, I show that $A$ has full rank, which is quite obvious but I really have difficulties to show that $B$ has full ...
2
votes
1answer
57 views
What does “the expansion of a vector relative to a basis is unique” mean in this proof of the Fundamental Theorem of Linear Algebra?
In the following proof, what does "But the expansion of a vector relative to a basis is unique" mean?
Let$$T:V\to W$$ be a linear transformation. Then $$dim(Im(T))+dim(ker(T))=dim(V)$$
$\mathbf{Proof:...
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1answer
12 views
Lower Dimensional Volume under Transformation
it is a well known fact that if $K$ is a measurable set in $R^n$ (we can restrict to convex bodies if you like), and $T$ a linear transformation then
$$|TK|=|detT||K|.$$
If $K$ is not $n-$dimensional ...
0
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1answer
13 views
Finite Rank Operator in Normed Space, not necessarily Hilbert neither Banach
Suppose that $E$ and $F$ are normed spaces and $T:E \rightarrow F$ is a bounded linear operator.
I NEED TO SHOW WHAT FOLLOWS:
If there are $n\in \mathbb{N}, f_{1}, ..., f_{n}\in E^{\ast}$ (dual of $...
1
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0answers
44 views
Given a Jordan canonical basis, how to find out to which generalized eigenspace picked generalized eigenvector belongs
Suppose we have finite-dimensional linear operator $A:V\to V$ , that has eigenvalues $\lambda_1 ,\lambda_2, ... \lambda_n$ . It is known that we can decompose $V$ into direct sum of generalized ...
1
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1answer
24 views
Which polynomial kernels are cyclic, and how to find a cyclic generator
Let $T$ be a $\Bbbk$-linear operator on a vector space $V$. Given $f\in \Bbbk[x]$ consider the operator $f(T)$ on $V$. Write $K(f)=\operatorname{Ker}f(T)$ and call such $T$-invariant subspaces ...
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1answer
23 views
If $T \colon V \to V$ is a linear operation and $B$ is a basis for $V$, then $T$ is one-to-one $\iff$ $[T]_B$ is invertible.
If $T \colon V \to V$ is a linear operation and $B$ is a basis for $V$, then $T$ is one-to-one $\iff$ $[T]$$B$ is invertible. Moreover if the equivalent statements hold, $[T$$-1$$]$$B$ $=$ $[T]$$-1$$B$...
1
vote
1answer
21 views
Find sum of Subspaces combination of constraints
Let $ S = \langle \{ x^2-1,x^2+1\} \rangle $
and $ T = \{p\in P_2(\mathbb{R}), p(x)= ax^2+bx+c : a+b=0,c=0\} $
$\color{black}{a) \ Find \ S+T}$
From the definition of Sum in subspaces, we take e.g $...
0
votes
1answer
32 views
Prove that BA cannot be invertible (A and B are linear transformations). Give an example of AB which is invertible.
The exact question I am trying to solve is as follows:
Let A: $\mathbb{R}^3\rightarrow\mathbb{R}^2$ and B: $\mathbb{R}^2\rightarrow\mathbb{R}^3$ be linear transformations, so BA: $\mathbb{R}^3\...
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1answer
69 views
How does the minimum distance between coordinates behave under change of basis?
$\renewcommand{\vec}[1]{\mathbf{#1}}$I came across this problem/idea quite some time ago and was unable to reach a conclusion. (The title of this questions may be misleading. I am open for suggestions....
0
votes
1answer
22 views
The set of all invertible operators need not be dense
I want to show that the set of all invertible operators $\mathcal{G}(\ell^2)$ is not dense in $\mathcal{B}(\ell^2)$.
Consider the right shift operator $T\in \mathcal{B}(\ell^2)$. We also know that $T\...
2
votes
1answer
40 views
Let $T \colon V \to W$ be a linear transformation, and if $\dim(V) < \dim(W)$, T cannot be onto.
Let $V$, $W$, be two finite dimensional vector space. Prove that if $T \colon V \to W$ is a linear transformation from $V$ to $W$ and $\dim(V) < \dim(W)$, then $T$ is not onto.
$$\dim(V) = \...
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2answers
33 views
Find the column space of the matrix $ \left(\begin{smallmatrix} 1 & 0 & -1 & 0 & 1\\ 0 & 1 & -1 & 2 &0\\ \end{smallmatrix}\right) $
I have the following matrix
$$
\begin{pmatrix}
1 & 0 & -1 & 0 & 1\\
0 & 1 & -1 & 2 &0\\
\end{pmatrix}
$$
And I am unsure as to how to write the column ...
1
vote
0answers
11 views
Vector orthogonalisation that keeps (as many as possible) zero elements
I have a matrix $\bf{A}$ of orthonormal vectors (columns). Most of these vectors have some elements which have a numerical value close to zero. For every column of $\bf{A}$ I go through all elements ...
1
vote
1answer
12 views
Basis for that $T: \mathbb{R}^3 \to \mathbb{R}^3$ is in rational canonical form
Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that
$T(x,y,z) = (x+y+z, x+y+z, x+y+z)$
Find a basis for $T$ such that your matrix$(A_T)$ is in rational canonical form.
I ...
0
votes
1answer
23 views
Does PCA always have to reduce dimensionality?
I came across this paper where the authors implement a regularized learning model to estimate the covariance matrix of a dataset. The authors say they "...propose a regularized form of Principal ...
3
votes
0answers
37 views
Proving row rank equals column rank of a matrix
I am studying Linear Algebra from the book of the same name by Larry Smith. I am confused with the way Smith proves that row rank of a matrix equals its column rank. Here are the things that he uses ...
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2answers
25 views
How to determine the matrix T to find the eigenvalues [closed]
For linear map $T : V → V$ , find the eigenvalues, and for each eigenvalue
λ find its algebraic and geometric multiplicities, and determine whether T is diagonalisable.
$V$ is the vector space of ...
0
votes
1answer
22 views
Prove that there exists an ordered basis $\gamma$ for which $[T^*]_\gamma$ has a column of $0$s.
$V$ is an $n$-dimensional vector space over a field $\mathbb{F}$. Assume that $T^*:V\rightarrow V$ is a linear operator on $V$ and $T^*$ is not an isomorphism. Prove that there exists an ordered basis ...
0
votes
2answers
35 views
prove for T:V→V is linear, if surjective then T is bijective
I know that $T$ is surjective means $R(T)=V$, which means $\mathrm{Nullity}(T)=0$.
But how can I show that $\mathrm{Nullity}(T)=0$ implies $T$ is injective?
I know that if $f(x)=f(y)$ and $x=y$ ...
0
votes
0answers
51 views
About derivative of delta function - chain rule for delta function containing a function
I have a problem that relates to derivative of a delta function. The problem originates from a paper I was reading https://aip.scitation.org/doi/full/10.1063/1.2938860
In the paper, it is said that ...
0
votes
0answers
10 views
Reflecting x-axis over a line of tangent of a function
I learned about linear transformations of vectors and shapes during the linear algebra class. Amongst all the transformations, I saw the reflections over x and y axis by multiplying certain matrices. ...
1
vote
1answer
19 views
2D Matrix Transformation Invariance
The transformation $\textbf{T}$ maps points $(x,y)$ of the plane into image points $(x', y')$ such that
$$\begin{align*} x' &= 4x + 2y + 14 \\ y' &= 2x + 7y + 42 \end{align*}$$
Find the ...
0
votes
2answers
45 views
How to find the rotation angle and axis of rotation of linear transformation?
I need some help with this problem:
We know that $T(x_1,x_2,x_3)=(x_2,x_3,x_1)$. We suspect that it is a rotation matrix, to be sure, we need to determine the rotation axis and the rotation angle.
...
0
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0answers
13 views
How do I use a set of orthogonal vectors as a new coordinate system?
I'm dealing with a problem where I need to apply a coordinate transformation to a set of data, but am not sure how. This is probably very simple, but I want to make sure I do it right.
Suppose that ...
0
votes
1answer
20 views
Coordinates to pixels
Let's say I have two geographical points $P_1=(512401.72N,0032120.17W)$ and $P_2=(512332.83N,0031948.64W)$. What would be the easiest way to scale line between the two so that it fit on a 800x800(...
0
votes
1answer
33 views
Show that $Γ$ is isomorphic to $L(V/W,V')$
Question: let $V$, $V'$ be vector spaces over field $K$ and $W$ be subspace of $V$ then show that $\Gamma = \{T\in L(V,V')\vert\forall w\in W: T(w) = 0\}$ is subspace of $L(V,V')$. Further show that ...
