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)
7,898
questions
2
votes
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 $\...
0
votes
1answer
17 views
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 ...
1
vote
2answers
24 views
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 ...
-1
votes
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 ...
0
votes
1answer
18 views
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 &...
1
vote
2answers
37 views
$\dim V \geq \dim U$, 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\...
-2
votes
0answers
12 views
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:
$$A_B^B=\begin{pmatrix}
1 & 1 & 0\\
0 & 1 & 0\\
0 & 0 & ...
0
votes
1answer
23 views
If I have a matrix in RREF is there a quick way of pulling out vectors which are not in the span of the row vectors of A?
Say I am looking for the kernel of a linear function $\mathcal{l}_A$: $\mathbb{R}^n $->$\mathbb{R}^m$ given by the $n \times m$ matrix A. Then say A is row-equivalent to some matrix B in RREF.
Is ...
0
votes
2answers
23 views
X subset of space vector V, $Tv=Sv, v \in X$ then T=S
Given two linear transformations:$$S,T: V \to W$$
such that $X\subseteq V$, and it is true that $S(v)=T(v), \forall v\in X$, and the exercise requests to prove that $S=T$. I have started by writing ...
2
votes
1answer
51 views
Confused about the relation between linear transformations, matrices and basis vectors
I was watching 3blue1brown's video series on linear algebra. My understanding till now is :-
A linear transformation takes in a vector and outputs another vector.
The above statement is equivalent to ...
0
votes
0answers
19 views
incluson between kernels for two linearly dependent operators. [closed]
Let $X$ be a Banach space. Let $A$ and $B$ be two bounded operators in $B(X)$, assume that $Ax$ and $Bx$ are linearly dependent for each $x \in X$ but $A$ and $B$ are linearly independent.
How we can ...
0
votes
0answers
30 views
When does a linear projection of a set of vectors preserve the span of the vectors?
Sorry if this question has been asked before, but I can't find it.
Suppose I have a set of vectors with some span. If I apply the same
linear projection $P$ to each vector in the set, the ...
-1
votes
1answer
25 views
Linear Algebra: an example of a linear operator [closed]
Is there a linear operator on $\mathbb{C}^4$ that is not diagonalizable? Can anyone give me an example please.
2
votes
1answer
26 views
Projection on $U$ along $W$ with $U,W$ contained in $V=U\oplus W$
Assume $V=U\oplus W$ is a direct sum where $U$ and $W$ are subspaces of the vector space $V$. Then we can define a linear map
$$E(v)=u,\\
with \qquad v=u+w\in V,\qquad u\in U,\ w\in W$$
Called the ...
2
votes
1answer
57 views
Riesz Representation Theorem geometric intuition
We just learned in our linear algebra class about the Riesz Representation Theorem, which states that if $V$ is finite-dimensional and $f$ is a linear functional on $V$, then there is a unique vector $...
0
votes
1answer
29 views
check if vectors are linearly independent?
Heyo, I'm just wondering if I'm correct in assuming that the following three vectors in four dimensional space are linearly dependent.. I simplified them using elemental row operations and ended up ...
0
votes
1answer
26 views
Unique differentiable linear operator mapping $\mathbb{R} \rightarrow \mathbb{R}^N$?
I am trying to find a mapping $\phi$ such that:
$\phi$ uniquely maps $\mathbb{R}$ to the subspace of $\mathbb{R}^N$ (for bounded $N$), where every dimension of the vector is bounded between $-1$ and $...
1
vote
3answers
18 views
Matrix of the differentiation operation [duplicate]
Exercise: Find the matrix of the derivative operation $D$ related to the base $\{1, t, t^2,..., t^n\}$ $$D: \mathcal P_{n} \to \mathcal P_{n}$$
I found a possible solution to this exercise, given that ...
1
vote
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. ...
2
votes
0answers
14 views
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 = ...
1
vote
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....
-1
votes
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'...
1
vote
3answers
23 views
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 ...
0
votes
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 ...
0
votes
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-...
1
vote
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 ...
1
vote
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:...
0
votes
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$ ...
-1
votes
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}$$
Which after some transformation $X$ then comes in the ...
0
votes
1answer
13 views
How to find Skew Projection Operator onto Plane parallel to some vector?
I was trying to solve previous year question paper of competitive exam
In that I observed some strange question which I have not encountered before.
They had given one equation of plane and told to ...
0
votes
1answer
23 views
For any two vectors $v_1$ and $v_2$, $\langle v_1, v_2 \rangle = \langle T (v_1), T (v_2) \rangle$ if $T$ is norm-preserving.
A linear transformation $T : V \to V$ is said to be norm-preserving for a given inner product if for all vectors $v$ in $V$ the following is true:
$\langle v,v \rangle = \langle T(v),T(v) \rangle$.
...
0
votes
0answers
15 views
How to find A such that $Ab$ is in the same direction of an eigenvector of $ACA^T$
Suppose $\mathbf{b}=[b_1,b_2]'$ is $2\times 1$ and $\mathbf{C}$ is a full-rank $2\times 2$ matrix which both are real and given. Now, consider the problem of finding a $2\times 2$ matrix $\mathbf{A}$ ...
1
vote
0answers
16 views
Prove that it's a Linear Functional (Kronecker delta)
Suppose that $dim(V) = n$ and that $B = \{e_1,...,e_n \}$ is a basis for $V$. Then we define $B^{*} = \{e^1,...,e^n \}$ such that:
\begin{equation}
e^{i}(e_{j}) = \delta_{j}^{i} = \begin{cases}1, ...
0
votes
1answer
48 views
Expectation of product of three or more dependent random variables
Given three random variables $X,Y,Z$, what is the formula of expectation of the product of three random variables? $$\mathbb{E}[XYZ]=?$$
Will it be less complex if we assume $X, Y$ and $Z$ as ...
0
votes
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,...
0
votes
0answers
19 views
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}) \...
0
votes
0answers
12 views
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 ...
0
votes
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}^...
0
votes
0answers
15 views
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 ...
1
vote
0answers
16 views
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.
...
0
votes
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'...
0
votes
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 ...
0
votes
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 ...
0
votes
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
votes
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 +(...
0
votes
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 ...
0
votes
2answers
32 views
Inequality involving ranks [duplicate]
I'm trying to prove the following inequality,
$$
\rho(AB) + \rho(BC) \le \rho(B) + \rho(ABC)
$$
where $A, B, C \in L(V)$, $V$ is a finite-dimensional vector space and $\rho(A)$ means the rank of ...
-2
votes
1answer
31 views
Prove that $T$ has at most two distinct eigenvalues [duplicate]
I need help with a problem in Axler's Linear Algebra text.
Any hint would be great.
Let $V$ be a vector space over $\mathbb{C}$ of dimension $n$, and $T: V\to V.$ If $$\dim \ker(T^{n-2}) \neq \dim \...
0
votes
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.
0
votes
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\\
\...
