Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...

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2answers
42 views

What's $\int \frac{1}{\sqrt{25-x^2}}$ [duplicate]

What is $$\int \frac{1}{\sqrt{25-x^2}}$$ WolframAlpha says $\sin^{-1}(\frac{x}{5})$ while I got $\frac{1}{5}\sin^{-1}(\frac{x}{5})$. What is correct? Thanks in advance.
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2answers
45 views

Prove that a continuous inverse-transformation of $f: [0,1) \cup \{ 2 \} \to [0,1]$ exists

I am having this transformation $f: [0,1) \cup \{ 2 \} \to [0,1]$ $$f(x) = \begin{cases} x & x \neq 2 \\1 & x = 2 \end{cases}$$ I've already proved that it is continuous. Question: Is ...
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1answer
15 views

Compute $\left(a_{i}A+B\right)^{-1},\qquad i=1,\ldots,N$ efficiently?

I need to compute the inverse matrix: $$(a_i A+B)^{-1}, \qquad i=1,\ldots,N$$ where $N$ is a large number. $A$ and $B$ are general $M\times M$ matrices independent of $i$. The only thing that ...
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0answers
15 views

Inverse transformation of continous transformation is bounded

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with $B \subseteq \mathbb C $ bounded. Now I want to proove that $ A = f^{-1} (B)$ is bounded! How can I proove that this ...
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0answers
31 views

Why aren't integration and differentiation inverses of each other?

Integration is supposed to be the inverse of differentiation, but the integral of the derivative is not equal to the derivative of the integral: $$\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\int ...
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1answer
19 views

inverse of a point $p$ respect to the circle $|z-z_0 |= r$ in complex

I was solving a problem to find the inverse of a point $p$ respect to the circle $|z-z_0|=r$. In my question I had to find inverse of $1+i$ w.r.t circle $|z+1-2i| = 2$. I applied the formula $q = z_0 ...
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1answer
42 views

Derivative of scalar function with respect to vector

Suppose I have three constant symmetric matrix $\mathbf{M}_{n\times n}$, $\mathbf{C}_{n\times n}$ and $\mathbf{D}_{n\times n}$ and two variable vectors $\mathbf{q}_{n\times 1}$ and ...
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1answer
23 views

Finding the inverse of a recursive function?

Let's say I have this function $$f(x) = \sum_{i=0}^{x-1}f(i)$$ provided $f(0) = 0, f(1) = 1$ and $x \in \mathbb Z$. This function is evidently one-to-one on $[3, \infty) $. Is there an inverse to this ...
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1answer
50 views

The invertibility of matrix $(I - XX')$?

$I$ is an identity matrix of size $n \times n$. $X$ is a matrix of size $n \times k$(Assuming $k \leq n$). As we know, $(I+XX')$ is invertible. Because $(I+XX') = (I(blank)X)*(I(blank)X)'$, where $(I ...
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1answer
58 views

Definition of inverse function

I have been wondering... Is there a mathematical equation for the inverse of a function? I mean apart from the typical "replace the x's with y's" way... I tried using the inverse function derivative ...
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1answer
39 views

How to simplify the inverse $(AB)^{-1}$ for rectangular $A$ and $B$?

Assume real rectangular matrices $A$ and $B$, where $A$ is $m \times n$, $B$ is $n \times m$, $m<n$, and the $m \times m$ product $AB$ is invertible. What are some possible strategies for ...
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2answers
33 views

Derivative Inverse of a function

I have a question: $\begin{array}{lrl} \mbox{If :} & f(x) & = x^5 + 3x^3 + 2x + 1 \\ \mbox{And :} & g(x) & = f^{-1} (x) \\ \mbox{What is :} & g'(7)&\mbox{?} \\ \mbox{What I ...
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3answers
46 views

How do you reverse $\frac{100n(n+1)}{2}=c$ to find n given c?

I'm developing a game where the character experience needed by level is given by Gauss' formula multiplied by 100: $ \dfrac{100\mathrm{level}(\mathrm {level}+1)}{2}$. So the experience table is ...
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0answers
12 views

Why is this Quadratic Form Independent of its Parameter in the Limit?

For $\alpha = e^{1/N}$, I have the following upper-triangular $\left(N+1\right)\times\left(N+1\right)$-Toeplitz matrix: $$\tilde{G}^{\left(N\right)}=\begin{pmatrix} 1/2 & 0 & 0 & \cdots ...
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1answer
13 views

$(T^{-1})^i == (T^i)^{-1}$?

I wonder if the hypothesis in the title is true. And if so, some ideas to prove it. I know $(A^T)^{-1} = (A^{-1})^T$ EDIT: Edited the title to match the generic answer. T does not have to be ...
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1answer
17 views

Prove that scalar functions of vectors cannot be inverted

The following seems obvious to me (because information is clearly lost), but I have no idea how to prove it: Suppose we have some arbitrary complex vector $\mathbf{A}$ with $m$ components. Let ...
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1answer
49 views

What is the correct $\det(A^{-1})$

Ok so I think I know why this is incorrect, because of the following: $$\det\frac{1}{ad-bc}\begin{bmatrix} d & -b\\ -c & a \end{bmatrix}\neq \frac{ad-bc}{ad-bc}$$ However, by adding a det ...
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1answer
70 views

If $A^2 = O$, is $A = O$?

I think the answer is "no", but I'm trying to find the flaw in this reasoning: $A^2 = O \implies AA = O \implies A^{-1}AA = A^{-1}O \implies A = O$ This shouldn't be true, as far as I know, so what ...
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2answers
31 views

Testing if a function has an inverse.

I was just wondering how you apply the rule: $$f(x_1) = f(x_2) => x_1 = x_2 $$ on the function: $$f(x) = x^3 - 9x^2 +33x +45$$ Any suggestions on how to proceed would be appreciated. EDIT: Yes ...
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1answer
34 views

Vertex Equation of an inverse quadratic function.

I'm working on a graphing web tool using JSXGraph, The user should be able to draw different functions. I was able to allow the user to draw quadratic functions by creating the vertex of the function ...
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3answers
23 views

Orthogonal matrices show that the product is also orthogonal

Show that if A and B are two orthogonal n × n matrices, then so is AB I know orthogonal is when the transpose of the matrix is equal to it's inverse. Please help
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1answer
29 views

For $A, B$ invertible matrices of the same order, is it true that $(A^T + B^T)^{-1} = (A^T)^{-1} + (B^T)^{-1}$?

If $A$ and $B$ are invertible matrices of the same order then is this statement true? Why? $$(A^T + B^T)^{-1} = (A^T)^{-1} + (B^T)^{-1}$$
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1answer
21 views

What kind of a matrix has a unitary diagonalizing matrix?

Suppose $D = P^{-1} A P$. When is $P$ unitary? In other words, what kind of a matrix $A$ should be, such that $D=P^{\dagger}AP$? i.e. what are the conditions a matrix must have to be able to ...
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2answers
21 views

Prove proposition on real numbers and inverses.

Prove the following proposition Let $x, y \in \mathbb{ R}>0$. If $x < y$ then $0 < y^{-1 }< x^{-1}.$ So far I've gotten that since $x, y > 0$ then $x^{-1}, y^{-1} > 0$.
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1answer
62 views

Linear Algebra - Find inverse of $A$

I have this problem : $$A = \left(\begin{array}{ccc} 3 & -1 & 1 \\ 2 & 0 & 1 \\ 1 & -1 & 2 \end{array}\right) $$ 1) Show that $A^3-5A^2+8A-4I=0$. 2) Using (1) To find ...
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1answer
49 views

What tools should be used to prove that a real function is one-to-one and onto?

Let $A = \mathbb R \setminus \{−1/2\}$ and $B =\mathbb R \setminus \{2\}$. Define $f : A \to B$ by the rule $$f(x) = \frac{4x − 3}{2x+1}$$ for all $x \in A$. Show that $f$ is one to one and onto. Find ...
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0answers
14 views

Solving simultaneous equations with matrices

I have a Matrix $B = \begin{pmatrix}2&1\\3&5\end{pmatrix}$ and its inverse $B^{-1}=\frac17\begin{pmatrix}5&-1\\-3&2\end{pmatrix}$ I also have a set of simultaneous equations: ...
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1answer
24 views

What properties do I have if I know $f$ and $f^{-1}$inverse are differentiable?

My goal is to show that $(f^{-1})'(y) = 1/[f'(f^{-1}(y)]$ for all $y$ in $(a,b)$. I have no idea where to start. I know that $f^{-1}$ and $f$ are differentiable.
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0answers
86 views

Inverse of $f(x) = xe^x-x$

I'm wondering if there is a way to obtain the inverse of the function $y=xe^x-x$. I am aware of the use of Lambert's W function in the inverse of $xe^x$ but as can be seen this is a different animal ...
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7answers
132 views

$f \circ g =\operatorname{ id}$ and $g \circ f \neq \operatorname{id}$?

Give two functions $f$ and $g$ s.t. $$f \circ g = \operatorname{id}$$ but $$g \circ f \neq \operatorname{id}$$ or a proof that this is impossible. This must be trivial, but I can't figure it out :) ...
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2answers
22 views

Showing that $\mathcal{G}(\ell_2)$ is not dense in $\mathcal{B}(\ell_2)$ via the right shift

This is my question: Is $\mathcal{G}(\ell_2)$ is dense in $\mathcal{B}(\ell_2)$? I am attempting to show that it is not by showing that the right-shift - call it $T:\ell_2 \rightarrow \ell_2$ - ...
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0answers
30 views

Inverse Relation of Irreflexive Property.

We are taking the inverse of relation to check that inverse of R is transitive, reflexive , symmetric and anti-symmetric to as it is on R (not inverse).. My question is that why we are not taking the ...
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0answers
26 views

Abscissa of absolute convergence of a Dirichlet series

I'd like some help to prove the following theorem : Let $\sum_{n \geq 1}\frac{f(n)}{n^s}$ and $\sum_{n \geq 1}\frac{g(n)}{n^s}$ be two Dirichlet series with respective abscissas of absolute ...
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1answer
45 views

Is this notation for inverse functions bad?

I'm trying to find useful notation for inverse functions that isn't too much in conflict with other notation already in use, but I'm wondering if this notation will come back and bite me in the ...
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0answers
19 views

Compute new inverse when old inverse and new and old matrix known

Say I have a matrix $M$ and know its inverse $M^{-1}$. Then every element changes so that $M'=M+(M'-M)$. Is there a fast way to find $M'^{-1}$ from this information? That is without computing the new ...
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1answer
26 views

The relation of domain and image of a function and its inverse

Theorem: Let both $f$ and $f^{-1}$ be functions. $\newcommand{\dom}{\operatorname{dom}}\newcommand{\im}{\operatorname{im}}$ Then $\dom(f) = \im(f^{-1})$ and $\dom(f^{-1}) = \im(f)$. Let $f: X ...
0
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1answer
17 views

Inverse of a function involving a Jacobian.

Why is it true that if the inverse of both $ \tilde{f} $ and $ f $ exists then: $$ \tilde{f}\left(\vec{x}\right) = [Df(x_{0})]^{-1} f(\vec{x}) $$ $$ \implies \tilde{f}^{-1}(\vec{x}) = ...
3
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1answer
40 views

The inverse of $(I-A)$ and the spectral radius of a nonnegative $A$ matrix

Suppost that $A$ is a nonnegative matrix, and let denote the identitiy matrix with $I$ and the spectral radius of $A$ with $\rho(A)$. Note that because $A$ is nonnegative according to the ...
0
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0answers
56 views

Formula for Area of parallelogram induced by linear operator

I'm given that the linear operator $L: \mathbb R^2\to\mathbb R^2$ is invertible. The set (u,v) is a linearly independent set in $\mathbb R^2$. I must find a formula for the area of the parallelogram ...
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1answer
20 views

How to verify an algebraic structure is a ring

I have a problem which ask me to verify that to structures are rings. However, I'm unsure of how exactly to check each property. I believe that the first is closed but not sure how to check the ...
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1answer
15 views

Linear maps, inverses and associated matrices?

This is likely a very simple question but if we have a linear map $f$ with an associated matrix $A$ is it a necessary and sufficient condition that for $f$ to have an inverse then $A$ must also have ...
3
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4answers
63 views

Finding inverse of a function $h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$

I have a function: $$h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$$ With just pen and paper, how can I determine if there exists an inverse function? Am I supposed to sketch it on paper to see if it can ...
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0answers
13 views

Using the inverse of the matrix find all the solutions of the following systems of equations?

I found the inverse using row operations and the identity matrix but I dont know where to go from here. Can someone direct me please ?
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1answer
25 views

Inverse matrix as a sum of matrix powers [duplicate]

I have matrix $ A\in \mathbb{C}^{n x n}$ and $A$ is invertible. How can I show that coefficients $c_0,...,c_{n-1}$ exist : $A^{-1} = c_0I+c_1A+...+c_{n-1}A^{n-1}$ I tried to solve it first by ...
2
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1answer
43 views

What is the Moore-Penrose pseudoinverse for a hermitian block-matrix with one zero block?

Given a block matrix of the form \begin{pmatrix} A & B^* \\ B & 0 \end{pmatrix} where $A$ is singular (otherwise one could simply use the well-known block matrix inverse), is there a ...
0
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1answer
34 views

Why are those equivalent transformations of inverse functions not the same thing?

Why are $\frac{1}{f}=\frac{1}{g}+\frac{1}{b}$ and $f=g+b$ not the same thing?
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0answers
24 views

Proofs with algebraic structures (rings)

If one is given a ring $R$ with a unity $u$, what are the steps one would have to take to prove that some element of $R$ named $s$ has a multiplicative inverse, where $-s$ also has a multiplicative ...
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1answer
19 views

Polynomial has right inverse implies invertible?

If $p:\Bbb R\rightarrow \Bbb R$ is a real polynomial such that $p$ has a right inverse $q$, does it follow that $p$ is invertible? That is, must $q$ also be a left inverse of $p$? The question ...
0
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1answer
21 views

Two roots of $\arcsin(x)$ in the range $[0,2 \pi]$

I am baffled with how to write the two roots of arcSin$(x)$ in the range $[0,2 \pi]$, while $x \in [-1,1]$, such that one root can be directly calculated in terms of the other root. For instance, we ...
0
votes
0answers
18 views

Compute the inverse fourier transform of $ e^{-Af} $ and $ e^{-A\sqrt{f}} $

I want to compute the inverse fourier transform of $ e^{-Af} $ and $ e^{-A\sqrt{f}} $, where $A$ is a constant and $f$ is frequency. In the case of $ e^{-Af} $, I tried to solve it from the Fourier ...