Transformation has many meanings in mathematics. If using this tag, add another tag related to the object being transformed. If there is a tag for your specific kind of transformations, use that one instead: e.g., (laplace-transform), (fourier-analysis), (z-transform), (integral-transforms), ...

learn more… | top users | synonyms

3
votes
0answers
133 views

Proof for a summation-procedure using the matrix of Eulerian numbers?

I've discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) ...
0
votes
3answers
80 views

A simple question on linear algebra and linear transformations

Let $f = (f_1, \cdots, f_m)$ be a function from $\mathbb{R}^n \to \mathbb{R}^m$. Prove that $f$ is linear if and only if for each $i$, $f_i$ is of the form $$f_i (x_1, \cdots, x_n) = a_1x_1 + ...
0
votes
1answer
87 views

How to transform this matrix & swap its columns?

I'm looking for a transformation matrix (or set of transformation matrices) that transforms matrix $\mathbf A = \begin{pmatrix} a&b&i&j\\ c&d&k&l \\ e&f&m&n \\ ...
1
vote
1answer
115 views

A simple question on linear transformations between vector spaces

Let $\phi: V \to W$ a linear transformation between vector spaces and $v_1, \cdots, v_k \in V$. Suppose that the following condition is fulfilled: $$\phi \left (\sum_{n=1}^{k}c_n\mathbf{v_n} \right) ...
3
votes
1answer
295 views

How to “flip” and change the sign of one particular row of this matrix?

I would like to transform the following matrix : $\mathbf A$ =$\ \begin{bmatrix} a&b\\ c&d\\ e&f\\ g&h \end{bmatrix}\ $ into this one : $\mathbf B$ = $\ \begin{bmatrix} g&-h\\ ...
1
vote
2answers
3k views

RYB and RGB color space conversion

I am working on a project where I need to convert colors defined in RGB (Red, Green, Blue) color space to RYB (Red Yellow Blue). I managed to solve converting a color from RYB to RGB space based on ...
0
votes
1answer
76 views

Define the linear transformation TA(v) = A(v) where (v) is the co-ordinate vector

I'm having some problems with this question. Could someone point me in the right direction? Thanks
0
votes
0answers
180 views

Linear transformation from P4 to M2x2

Can someone verify the formula in the last line. It seems that the LHS is not equal to the RHS The linear transformation of P4 ---> M2x2 is given by $$ T(ax^4+bx^3+cx^2+dx+e) = $$ $$ ...
0
votes
2answers
46 views

An explanation about terminology in vector spaces

Call a linear transformation $\rho: V \to V$ ($V$ is a vector space) idempotent if $\rho^2 = \rho$. Prove that if $\rho$ is idempotent, then it acts as the identity on $\rho(V)$. If I understand the ...
2
votes
2answers
86 views

Meaning of $p(\phi)$ where $\phi (x,y) = (x+y, x- 2y)$ and $p(x) = x^2 -2x + 1$

Consider the linear transformation $\phi : \mathbb{R}^2 \to \mathbb{R}^2$ defined by $\phi (x,y) = (x+y, x- 2y)$. Let $p(x) = x^2 -2x + 1$. Does $p(\phi)$ make sense and if yes what is it?
0
votes
1answer
23 views

Rescale a function

Lets assume that $A_1$ a function with maximal value $M$ and minimal value $m$ with $M \neq m $ How can i find the transformation that maps this function to $A_2$ with $N$ as new maximal value and ...
1
vote
1answer
438 views

Are the following transformations linear?

I'm preparing for my exam and I am stuck at these two exercises in which I must prove that the given transformatios are linear. I know that a transformation is linear, if it's closed under adition and ...
3
votes
3answers
2k views

Find the spanning set of the range of the linear transformation $T(x)=Ax$.

Let $$ A= \begin{bmatrix} -4 & -4 & 12 & 0 \\ -4 & -4 & 12 & 0 \\ 4 & -2 & 0 &-6 \\ 1 &-4 &7 &-5 \\ ...
0
votes
1answer
262 views

Standard Basis of the Finite Field of Prime Numbers

A little info regarding this field: Addition and multiplication in $Z^n_p$ behave as usual but with the remainder taken upon division by $p$. Ex: $Z_3$ will only consist of the three integers ...
1
vote
1answer
220 views

How to convert a sequence of numbers in the formula?

I'm trying to understand different sorting algorithms and their BigO notation. Suppose, I'm using insertion sort and I have the worst case: 6 | 5 | 4 | 3 | 2 | 1 ...
3
votes
1answer
110 views

Question about special orthogonal Lie group construction

Working through homework and I run into this problem: Suppose the Lie group $SO^{+}(2,2)$ is presented as the group of all transformations in its associated space. How do you determine whether a ...
1
vote
2answers
2k views

Diff eq. transformation polar coordinates

I have $(x',y')=(x-y-x(x^2+y^2)+\frac{xy}{\sqrt{x^2+y^2}},x+y-y(x^2+y^2)-\frac{x^2}{\sqrt{x^2+y^2}} )$ Now I want to use polar coordinates $(x,y)=(r\cos(t),r\sin(t))$ to get ...
0
votes
0answers
461 views

Graph Transformations Order of Operations

I'm a little confused regarding order of operations for shifted graph functions. For part b, which one is correct, and why? Shifting two to the left, and flipped over the x-axis, or flipped over ...
1
vote
1answer
125 views

Householder reflections

Let $x=\begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}$ I want to use a Householder reflector U to keep only first element in vector x, and make everything else zero but I'm doing something wrong... ...
1
vote
0answers
55 views

distance to sheared right circular frustum

How do I calculate the distance to a sheared right-circular frustum? In particular, I'm shearing in a direction perpendicular to the axis, so the cross sections remain parallel circles. I know I can ...
2
votes
1answer
724 views

Causal Inverse Z-Transform of Fibonacci

Say the Fibonacci sequence is defined by: $y(n) = y(n-1) + y(n-2)$ initial conditions: $y(0)=0, y(1)=1$ I incorporate those initial conditions as: $y(n) = y(n-1) + y(n-2) + \delta(n-1)$ ...
1
vote
0answers
132 views

question about coordinate transformation

In isoparameteric finite element of second order tetrahedron element, the original coordinate $(x,y,z)$ would be transformed to standard coordinate $(\xi, \eta, \zeta)$ by a shape function. As a ...
0
votes
0answers
47 views

Further Properties of Adjoints

I recently posted a question here about adjoint maps. I now want to take this further and show that - with the assumptions as in the original - the minimum polynomials of $T$ and $T^{*}$ are the same. ...
2
votes
1answer
110 views

Properties of Adjoints

Suppose we have a linear transformation $ T $ on some real inner product space $ V $, with adjoint $ T^{*} $. How can we go about showing that $$ (T^{n})^{*} = (T^{*})^{n} $$ for a positive integer ...
3
votes
3answers
960 views

Rank of matrix AB when A and B have full rank

Define $A$ as $m\times n$ matrix with rank $n$, and $B$ as $n\times p$ matrix with rank $p$. Calculate the rank of matrix $C=AB$. --edit-- Rank of a matrix is the number of linear independent rows.
3
votes
1answer
132 views

If $AB=0$, then $A+A^T$ or $B+B^T$ is singular

Define $A$ and $B$ as being square matrices of dimension $2011$. Prove that if $AB=0$, then at least one of matrices $A+A^{T}$ or $B+B^{T}$ have rank below $2011$. -- edit -- Rank of a matrix is ...
1
vote
1answer
107 views

Region of convergence of Z-Transform connected area?

Shouldn't the Region of Convergence of the Z transform be a connected area ? In Oppenheim solution manual, I've found this answer of a question that asks to determine the different forms of the ...
2
votes
2answers
178 views

Finding if the equation is even or odd

I am learning fourier transform and I came across this question in which author right away says the given equation is "even". How does this equation become "even"? $$x[n]=\begin{cases}A & -M\le ...
2
votes
1answer
208 views

Fourier Transform of an Operator

I need to calculate the fourier transform of an Operator. meaning I need to calculate the transform of the Operator's corresponding convolution kernel. so the question is: 1.given a 2d fourier ...
1
vote
1answer
256 views

How to deal with non random data in statistical analysis?

I have a set of monthly water quality data, and I want to use them in a few statistical analysis (such as finding distribution or using in copula models) which require random variables as input. I ...
4
votes
3answers
486 views

Image of function definition notation

In my Linear Algebra and Geometry textbook, it defines the image of a linear transformation $T$ as: $$\operatorname{Im}\, (T) := \{\; w \in W : \; w=Tv \;\;\text{ for some } v \in V \} $$ As far as ...
1
vote
0answers
104 views

what's the difference between Cohen-Daubechies-Fauraue 9/7 transformation and Discrete cosine transform?

can someone explain, what's the difference between this 2 tranformations? DCT and CDF Greetings
1
vote
0answers
252 views

Multiple integral variable substitution using Jacobian matrix and matrix rotations

Question: By an appropriate choice of new variables evaluate the integral $\int\int_R(x^2+y^2)\,dx\,dy$ over the interior of the square bounded by $y=\pm x$ and $y=\pm (x-2)$. I sketched the square ...
5
votes
3answers
174 views

Find the necessary and sufficient conditions on $A$ such that $\|T(\vec{x})\|=|\det A|\cdot\|\vec{x}\|$ for all $\vec{x}$.

Consider the mapping $T:\mathbb{R}^n\mapsto\mathbb{R}^n$ defined by $T(\vec{x})=A\vec{x}$ where $A$ is a $n\times n$ matrix. Find the necessary and sufficient conditions on $A$ such that ...
1
vote
2answers
893 views

linear transformation and angles?

Does linear transformation, prevent preserve angle between two vectors? I guess that it is true, so if we translate normal vector of a plan, it will be orthogonal to translated plan.
3
votes
2answers
574 views

Kernel density estimation for heavy-tailed distributions using the champernowne transformation

I am trying to follow this paper to estimate the density for a heavy-tailed distributions using the champernowne transformation. Alternative link to the paper Another alternative link to the paper ...
0
votes
1answer
65 views

How to come up with a formula for converting a set of numbers?

I have a set of values (W1, X1 and T1). Via some magic, the combination of these numbers result in a number X2. The full spreadsheet is here, but the gist of the problem looks like this: ...
5
votes
2answers
909 views

Why is the absolute value needed with the scaling property of fourier tranforms?

I understand how to prove the scaling property of Fourier Transforms, except the use of the absolute value: If I transform $f(at)$ then I get $F\{f(at)\}(w) = \int f(at) e^{-jwt} dt$ where I can ...
1
vote
2answers
61 views

Given a projection T, how do we prove that T(v) = v in this situation?

Let $V$ be a finite dimensional vector space and let $T: V\to V$ be a linear transformation. Assume that $T$ is a projection - i.e., $T^2(v) = T(v)$ for every $v \in V$. Assuming that $v \in ...
1
vote
1answer
414 views

Geometric transformation on circle equation

Suppose that I have variables $x_1,x_2$ and following circle equation: $x_1^2+x_2^2=1$. Now I have a matrix $A$ which will be used to transform my circle equation. $Z=AX$ where $X$ is a vector with ...
2
votes
1answer
231 views

Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three ...
3
votes
1answer
74 views

Scale change on Quadratic

Consider the two functions $f(x)=ax^2$ and $g(x)=bx^2$. Using this transformation form $T(x,y)=(cx,cy)$, find a scale change that maps $f(x)$ onto $g(x)$
0
votes
0answers
40 views

R to $R^{20}$ linear transformation for classification?

I try to do some type of classification.I have 2 timeseries signals to distinguish from eachother. Here is the way i was told to do it: I take into account one of these signals and call it MySignal ...
1
vote
1answer
2k views

Matrix representation of a transformation in a basis $B$

I need some clarification on this problem; my class notes and my current thought process are conflicting. I have a linear transformation $$T(a,b) = (a+2b, 3a-b)$$ and I'm asked to find $[T]_B$ ...
1
vote
1answer
3k views

Find the Matrix A of the Linear Transformation

Can anyone walk me through the steps to complete this problem? I am unsure of where to start to solve the problem. I get that the resulting matrix $A$ should be a $2 \times2$ matrix, should I be ...
0
votes
2answers
440 views

Relationship between null space and invertibility of linear transformations

Is there a relationship between the null space $N(T)$ of a linear transformation $T$ and whether or not it is invertible? For example, if you know $N(T) \neq \{0\}$, can you be sure it's not an ...
1
vote
3answers
713 views

Is this an invertible linear transformation?

Suppose you have a linear transformation $T: M_{2\times 2}\to M_{2\times 2}$ given by $$ \begin{pmatrix} a & b \\ c & d\end{pmatrix}\mapsto \begin{pmatrix} a+b & a \\ c & ...
1
vote
2answers
122 views

Whether linear transformation maps $0$ to $0$

Suppose I have a linear transformation $T: V\to W$. If I perform this transformation on the $0$ vector of $V$, $0_V$, does that necessarily mean its image will be $0_W$? In other words, is it ...
1
vote
1answer
659 views

Find the Matrix of a Linear Transformation.

It's been a few weeks since the subject was covered in my Linear Algebra class, and unfortunately linear transformations are my weak spot, so could anyone explain the steps to solve this problem? ...
0
votes
3answers
649 views

Composite Linear Transformations

Give an explicit example of a pair of linear transformations $T : V \to W$ and $S : W \to U$ between vector spaces $V$, $W$, and $U$, so that neither $T$ nor $S$ is the zero linear transformation, but ...