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1answer
364 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 ...
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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 \\ ...
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2answers
225 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
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1answer
153 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
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1answer
106 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
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2answers
1k 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 ...
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0answers
439 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
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1answer
113 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... ...
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0answers
50 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
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1answer
585 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: $y(n) = y(n-1) + y(n-2) + \delta(n-1)$ ...
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0answers
129 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 ...
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0answers
45 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
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1answer
97 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
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3answers
598 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
131 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
95 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
166 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
177 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
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1answer
226 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
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3answers
385 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 ...
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0answers
99 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
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0answers
229 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
170 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 ...
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2answers
671 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
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2answers
494 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
60 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
648 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
59 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
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1answer
300 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
203 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
68 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 $g(x)$
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0answers
38 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
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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
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1answer
2k 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 ...
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2answers
285 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
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3answers
558 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
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2answers
121 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
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1answer
555 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
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3answers
517 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 ...
1
vote
1answer
2k views

Negative & Positive Shear Factor

My question relates to constructional geometry & matrices aren't to be involved in the solution because stated Math level is up to O Levels... The figure below shows shear with y=3 as invariant ...
0
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2answers
75 views

Number mapping function

I can't find out a function f(x)=y that would map my x's to required y's. It is OK to write it in a programming language. Notation in mathematics is also OK. It ...
3
votes
1answer
539 views

Translation Matrix and Why non linear?

When we translate a point $p_3 = (x,y,z)$ to coordinates $p_4 =(x + t_x , y + t_y ,z + t_z,1)$ we use $4 \times 4$ Translation matrix using homogenous coordinates, hence we add a $1$ to fourth ...
1
vote
1answer
146 views

Derive Rigid Transform Matrix from Axes and Origin

I'm trying to derive the matrix of a rigid transform to map between two coordinate spaces. I have the origin and the axis directions of the target coordinate space in terms of the known coordinate ...
1
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1answer
112 views

Discover matrix of linear transformation between non-canonical bases

I'm solving a linear algebra problem. I have linear transformation $D$: $D : R_2[t] \rightarrow R_2[t]$ $D(p) = \frac{d}{dt}p$ and bases: $A = \{1 + t, 1- t, t^2\}$ $B = \{1 + t, 1 - t\}$ Now I ...
1
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1answer
70 views

Correlated Equilibrium - Transforming a non-linear objective function into a linear one

I am trying to transform a non-linear objective function into a linear one, in order to create a LP. How might I go about to do this (I have never taken a course in linear programming). I have that I ...
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3answers
242 views

$f: V \mapsto V$ is a projection, show $V = \ker(\pi) \oplus \operatorname{Im}(\pi)$ [duplicate]

Possible Duplicate: Show that $V = \mbox{ker}(f) \oplus \mbox{im}(f)$ for a linear map with $f \circ f = f$ I think I need to use the fact that if $v \in V$, then $v = (v - \pi(v)) + ...
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3answers
249 views

Pre–emphasis - Signal Processing

I am trying to compute the Pre-emphasis of a signal and the formular is below: y[n] = x[n] - 0.95 x[n-1] Let: ...
2
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1answer
57 views

Need an explanation of a particular expression transformation

Please, I need an explanation of the one transformation. I have the equation set and its solution. $$ \begin{cases} \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 3\\\\ \frac{y}{x} + \frac{z}{y} + ...
0
votes
0answers
260 views

Girsanov Transformation Example

Is this the correct use of Girsanov's transformation where $B_{n}$ is a discrete Brownian motion? For example computing: $E[(B_{n}+2n)^{2}]$ Set: $\widetilde{B_{n}}=B_{n}+2n$ And ...
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0answers
17 views

Need help transforming an list of numbers into some uniform list in order to apply the rule mentioned inside more effectively

So I have a list of values like so L1 = [-4 -3 5 8 ]; Note the sum of each element in L1 will always be > 0. The operation I am performing on L1 is as follows ...