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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Diagonalization of a matrix by a rotation matrix

Given is the following matrix $$A := \begin{pmatrix} 3x & 4x & 0 \\ 4x & 3x & 0 \\ 0 & 0 & 5x \end{pmatrix} $$ whereas $x \in \mathbb{R}$ is a constant. This matrix can be ...
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0answers
39 views

[Linear Algebra][Recursive Sequence] A general function form from recursive sequence

I'm searching method to create a general form of function from recursive sequence. I've tried to use the linear fractional transformation, however I don't know whether this is a good idea to ...
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0answers
30 views

Determine the formula for the linear transformation $T: \mathcal{R}^3 \rightarrow \mathcal{R}^3$ [duplicate]

Consider the subspace $\mathcal{U}$ = span{$\mathbf{u}_1, \mathbf{u}_2 \}$ of $\mathbb{R}^3$, where $ \mathbf{u}_1 = \left[\begin{matrix} 3 \\ 2 \\ 6 \\ \end{matrix}\right] \text{ ,} \mathbf{u}_2 = \...
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1answer
52 views

Reference/Info Request on this Alternate Method of Calculating Eigenvectors

I would like to request resources where I can find more examples and possibly a proof of the following method/algorithm of calculating eigenvectors. I came across it during a seminar I attended, but I ...
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0answers
5 views

When are the singular functions in the singular system of a compact operator in L2 bounded in sup norm?

Suppose I have some compact linear operator $A: L_2 (\mu_1)\to L_2(\mu_2)$ where $\mu_1$ and $\mu_2$ are probability measures on some subset of Euclidean space. Such an operator admits a singular ...
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1answer
30 views

Scale weighted variable

I would like to scale a vector $x$ such that after weighting, it has the following properties: $\sum_{i} w_{i} x_{i} = 0$ and $\sum_{i} w_{i} x_{i}^2 = 1$ where $w_{i}$ are weights that add up to ...
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0answers
12 views

How to scale a weighted variable for linear regression?

I would like to scale a variable such that after weighting it is scaled to mean zero and standard deviation one: $\sum_{i} w_{i} x_{i} = 0$ and $\sum_{i} w_{i} x_{i}^2 = 1$ where $w_{i}$ are ...
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1answer
27 views

Dimension of a vector space of endomorphisms

I'd like to get some help in a type of problem I'm a little bit unfamiliar with, which is finding basis for function vector spaces Let $V$ be the space of endomorphisms of $\mathbb{R}^{3}$, then we ...
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2answers
43 views

Find the image and kernel of the linear transformation.

Find the image and kernel of the linear transformation. $T: P_{1}\rightarrow P_{2} , T(p(x))=xp(x)+p(0)$? I'm trying to solve this exercise, but I'm stuck with the linear transformation notation, I ...
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1answer
41 views

Existence of Unique Linear Transformation implies Basis

Here is a problem that's bugging me: Let $V$ be a vector space over a field $F$ and let $v_1, ..., v_n \in V$. Prove the following: If, for any vector space $W$ over $F$ and any elements $...
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2answers
44 views

Obtaining the matrix representation of a linear transformation [closed]

My question is about the answer here. Here, in this answer why: $$ Av_j = Ae_j. $$
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0answers
26 views

Translation as product of reflections

I am facing the following problem, given the translation in the euclidean affine space of dimension 4 $ \tau_v= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ ...
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1answer
43 views

To find minimal polynomial of a linear transformation

Let $V$ and $W$ be finite dimensional vector spaces over $\mathbb{R}$ and let $T_{1} : V \rightarrow V$ and $T_{2} : W \rightarrow W$ be linear transformations whose minimal polynomials are given by $...
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1answer
45 views

Numerical analysis of the terms of the Volterra series of high orders [closed]

The Volterra series contains a number of terms of order 2 and above. Each of the addends is a generalization of the convolution integral of 2 and higher output signals with 2 and higher output signals,...
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0answers
26 views

Composition of transformations of a plane

How do I find a composition of linear transformations $$T_a\circ R_A^α\circ T_b$$?
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0answers
53 views

Viewing the columns of a matrix as vectors, why do row operations not change the determinant?

Viewing the determinant as the amount of space enclosed by the columns of a matrix, it's geometrically clear to see why column operations don't change the determinant. All adding one column to ...
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2answers
43 views

Linear algebra- matrix of rank $r$ as product of two matrices of rank $r$

Let $A$ be $m\times n$ matrix of rank $r$ then show that $A$ can be written as product of matrices of order $m\times r$ and $r\times n$. I know how to prove the above relation if $r=1$ but don't know ...
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0answers
120 views

How exactly does Hahn-Banach theorem explain duality of vector spaces?

Serge Lang's Linear Algebra textbook just introduced me to the concept of dual space in very formal terms: space of all functional transformations having co-domain as $1$-dimensional vector space over ...
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1answer
26 views

Does intersection of invariant subspace distribute over direct sum of invariant subspace found by minimal polynomial?

Let $T: V\to V$, and $V$ is of finite dimension. $M$ is the minimal polynomial $M = M_1 \cdot M_2 \cdot\dots\cdot M_k$. $M_i$ monic polynomial. $V = W_1 \oplus W_2 \oplus \dots \oplus W_k$. It's ...
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0answers
16 views

Existence of Annihilating Polynomial for a Linear Operator over a fine dimensional vector space.

How do I show the existence of Annihilating Polynomial for a Linear Operator $T:V\to V$ over a finite dimensional vector space $V$. I had an idea that $\mathbb{L}(V,V)$ is a vector space of dimension ...
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1answer
41 views

Why is orthogonal projection not always multiplication by a diagonal matrix?

Suppose I have a vector $v$ which I want to orthogonally project onto a subspace $S.$ The subspace is defined as the space spanned by the columns of a matrix $A.$ I could do this by constructing a ...
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1answer
58 views

Insight on the polar decomposition of a shear?

I recently learned it, and really love the polar decomposition of a matrix, because it was the first time I actually could picture what it meant to "apply a transformation to space" (a phrase I kept ...
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3answers
38 views

Number of Linear Transformations having same Kernel space dimensions

Let T:V->W. Is it possible to have multiple T's that have the same dim(ker(T)) dimensions of the kernel of T ? If not why ?
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1answer
51 views

Number of zeros in $A^k$

Given a nilpotent matrix $N$ of order $n\times n$ such that $N^k=0$. Then i know that $m_{N}(x)=x^k$ and $c_{N}(x)=x^n$. This means that $N^k$ is a zero matrix. My question now is can we deduce how ...
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1answer
28 views

Self Adjoint and Skew Adjoint Linear Transformations

I'm studying adjoints, and I'm confused as to how I prove this. I have a definition of a self-adjoint $T$, such that $T^*=T$, where $T^*$ is the adjoint. I then have that the definition of a skew-...
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2answers
25 views

Finding the rigid body transformation - how many points are required?

Apologies for a perhaps very stupid question, but I've begun to confuse myself a little, I think. If I have two sets of 3D points which are related by a transformation matrix, how many corresponding ...
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1answer
33 views

Self-adjoint operator is diagonalisable

I am revising adjoints for a linear algebra exam and am confused as to how to prove this. Suppose that $T: V \rightarrow V$ has the property that $T^*=aT$ for some complex a. How then do you prove ...
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2answers
50 views

Kernel of a polynomial which divides the minimal polynomial

Question: $P_{T}$:= characteristic polynomial of T; $minP_T$:= minimal polynomial of T. Consider $P_{T}=minP_T$. If $g$ is a polynomial which divides $minP_T$, then dim(Ker($g(T)$))=deg $g$. My ...
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2answers
41 views

PMA Rudin: understanding the definition of matrices, definition 9.9

I have background of linear algebra but am still confused about the definition. Suppose $\{\mathbf{x_1}, \cdots, \mathbf{x_n}\}$ and $\{\mathbf{y_1}, \cdots, \mathbf{y_m}\}$ are bases of vector ...
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0answers
27 views

Find a basis and coordinates for a second degree polynomial

Find a basis $B$ for $P_2$ $[p]_B = \begin{bmatrix}p(0)\\p(1)\\p(2)\end{bmatrix}$ and its coordinates to a second degree polynomial The solutions says: $p(x) = p(0)e_1(x) + p(1)e_2(x) + p(2)e_3(x)...
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1answer
23 views

At least one eigenvalue among all roots

Let $V$ be a vector space and $f \in \text{End}V$. Let $p$ be a polynomial over a field $K$ so that $p(f)=0$. Also we have deg $p = m$ and $c_1,c_2,\dots,c_m$ be all roots of $p$. Prove that at least ...
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2answers
50 views

Prove that if the minimum polynomial is the product of distinct factors then the transformation is diagonalizble.

I know that If $f$ is diagonalisable then its minimal polynomial is the product of distinct linear factors. Now, how to prove the converse. That is: Let $A:E\to E$ be a linear transformation from ...
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0answers
31 views

Determinant of Similar Matrices vs Matrix of Change of Basis

I am little confused when we say that all similar matrices have same determinant. The Proof goes something like: All similar matrices can be represented as $$B=X^{-1}AX$$ hence $det(B) = det(X^{-1}AX) ...
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1answer
84 views

Prove {$I$ , $A$ , $A^2$ ,…, $A^{n-1}$} is a basis for W

Let $F$ be a field , $A$ $\in$ $M_{n\times n} $($F$) (the set of all $n\times n$ matrices over field $F$ ) and $W$ = { $ B \in M_{n\times n } (F)$ $|$ $AB = BA$ }. Suppose there exists a column ...
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0answers
14 views

I want to find an explicit direct sum decomposition of the space into T-cyclic subspaces

I construct a linear operator $T \in \mathcal{L(\mathbf{C}^7)}$ , where the minimal polynomial is $m_T (x) = x^2(x-1)^2$ and the caraterisitic polynomial is $p_T (x) = x^3(x - 1)^4$. The linear ...
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2answers
26 views

If each complex operator has invariant subspaces of any dimension?

I know that each operator in a $n$-dimensional complex space $A:E\to E$ is traingularizble. That is there exists a basis of the eigenvectors of $A$ wit which $A$ is triangular. Now this claim would ...
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0answers
35 views

If for linear operator $A:E\to E$, $A^k=0$, then $A^n=0$ [duplicate]

Let for the linear operator $A:E\to E$, $A^k=0$, then $A^n=0$. Here $n=dim E$ and $n <k.$ I tried to find the characteristic oolynomial that I think it is the clue, however did nit succeed. Any ...
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2answers
36 views

If $L: V\to V$ is a linear operator, and $V$ has bases of $n$ vectors, can we say rank$L = n$?

If $L: V\to V$ is a linear mapping, and $\{v_1, ..., v_n\}$ and $\{L(v_1), ..., L(v_n)\}$ are bases for $V$, can we say that rank$L = n$ ? I'm trying to prove that nullity$(L) = 0$, so my thought ...
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0answers
55 views

Are all isometries of subsets affine?

I've found at least two questions that deal with whether isometries are affine, Are isometries always linear? and Should isometries be linear? However, both of these questions assume we are ...
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1answer
20 views

How to create a transformation matrix for a M22 → M22 transformation

I have a linear transformation, T, such that; T:${M_{22}}$→${M_{22}}$: T$\left(\begin{bmatrix}{x_{11}} & {x_{12}}\\{x_{21}} & {x_{22}}\end{bmatrix} \right)= \begin{bmatrix}{{x_{12}}-5{x_{21}...
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0answers
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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 ...
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1answer
39 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: $\...
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1answer
15 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 ...
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0answers
22 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
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1answer
127 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: ...
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1answer
15 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 ...
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0answers
18 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 ...
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0answers
25 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 ...
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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
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2answers
50 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}\...