0
votes
1answer
48 views

How to solve for $f(2)$ given $3f(x)+f(2-x) = 2x^2$?

I just came across the problem where I'm given that $3f(x)+f(2-x) = 2x^2$ and I need to find $f(2)$. I know it is simple, please help me.
3
votes
2answers
83 views

Manipulating identities

I'm having some trouble deriving certain identities. If $$S(z) = \prod_{i=1}^n (z-z_i)$$ then how can I write $$\frac{1}{S(z)}\frac{d^2S}{dz^2} = \sum_{i=1}^n\frac{1}{z-z_i}\sum_{j\neq ...
2
votes
2answers
31 views

How to see a function as a vector in a vector space

I know that strictly speaking my question is some sort of a duplicate of at least this previous one and I am quite sorry for that (usually I try to get the best from previous questions), but still I ...
0
votes
0answers
33 views

Scaling a big range of small numbers to a small range of big numbers

I'm trying to make a volume meter in a Flash program. I have data coming in like: 0.008 0.0005 0.1 0.02 These numbers indicate the volume of a sound coming in ...
1
vote
1answer
29 views

Proving properties of linear maps on one-dimensional vectors

An exercise from the book "Linear Algebra Done Right" asks to prove the following: 'Show that every linear map from a one-dimensional vector space to itself is multiplication by some scalar. More ...
0
votes
0answers
30 views

Quick Question on showing a function is an inner product

I just have a quick question How come =p(1)q(1)+p(2)q(2) is an inner product but =p(1)q(1)+p(2)q(2)-p(3)q(3) is not?
0
votes
0answers
37 views

Proof of two properties of a simple math function

I would like to define a function to evaluate the value for some entities which receive a number of "up"s ($\mathcal{u}$) and "down"s ($\mathcal{d}$). I devised the following function: ...
0
votes
0answers
28 views

How to define conditions under which linear maps are injective?

In this book (http://linear.axler.net/) proposition 3.2 states the following: Proposition 3.2: A linear map $T : V \rightarrow W$ from vector space $V$ to vector space $W$ is injective if and only if ...
0
votes
0answers
24 views

How to calculate the inverse of a known optical distortion function?

Assume I have the following lens distortion function: $$ x' = x(1 + k_1r^2 + k_2r^4) \\ y' = x(1 + k_1r^2 + k_2r^4) $$ where $r^2=x^2 + y^2$. Given the coefficients $k_1$ and $k_2$ I would need to ...
0
votes
4answers
71 views

Why is this not a 1-1 Function?

Linear mapping: \begin{align*} F: \mathbb R^3 &\to \mathbb R^2,\\ \begin{pmatrix} x\\y\\ z \end{pmatrix} &\mapsto \begin{pmatrix} x\\y\\ \end{pmatrix} \end{align*} I thought the check for ...
0
votes
1answer
51 views

Infinite dimensional operator inverse

A is a linear operator on V and there exist a single operator B on V such that AB = I or BA = I. Prove that then A is monomorfic and epimorfic. On infinite dimensions, left and right inverses need ...
1
vote
0answers
21 views

Mathematical standard term for a function of (different) operator arguments

In quantum mechanics, one often considers functions of linear operators, like $$f(A,B) = A\cdot B + e^A \cdot B^2$$ where $A,B$ are linear operators. In physics this often causes confusions, as some ...
0
votes
3answers
61 views

Function Notation question that needs an answer

$f(x)= f(x+1)+3$ and $f(2)= 5$, determine the value of $f(8)$. I don't understand how $f(x)$ can equal $f(x+1)+3$
0
votes
0answers
24 views

Convert function form to variate polynomials form

Let $f:{\Bbb Z}\longrightarrow{\Bbb Z}$ be a function, example $f(0)=3,\ f(1)=2,\ f(2)=0,\ f(3)=1$. Now that was "in decimal". We can think of it "in binary" format: $f(00)=11,\ f(01)=10,\ f(10)=00,\ ...
1
vote
1answer
36 views

Proving Linear Independence of Gaussian Functions

Assume that I have a summation of $N$ Gaussian functions with different means $\mu_i$, $1 \leq i \leq N$ as in $$ \sum_{i=1}^{N} a_i e^{ - (x- \mu_i)^2 }, $$ where the $a_i$ are real numbers. Is it ...
0
votes
2answers
50 views

degree of homogeneity

I have the function $$f(x,y)=\frac{y^b}{x^a}+\frac{x^b}{y^a}\quad a,b\gt0$$ The questions I have to answer are For which a and b is the function homogenous? Determine the degree of homogeneity My ...
1
vote
2answers
26 views

Functions That Retain Their Form When Inverted

Are linear functions the only functions that retain their form when inverted -i.e. an exponential function becomes a log function when inverted, a square function becomes a square root when inverted, ...
1
vote
1answer
52 views

Set of Functions is a Vector Space problem

Let $F$ be a field. Consider the $F$-vector space $F(X, F)$ of all functions from $X = \{ 1, 2 \} \to F$. Define $e_1, e_2 \in F(X, F)$ by $e_1 = \{ (1, 1),(2, 0) \}$ and $e_2 = \{ (1, 0),(2, 1) \}$. ...
0
votes
1answer
44 views

Characterization of linear functions in $\mathbb{R}$ using distance

First of all, by a linear function in $\mathbb{R}$, I mean a function $f:\mathbb{R}\rightarrow\mathbb{R}$ of the form $f(x)=ax+b\ \forall x\in\mathbb{R}$, where $a,b\in\mathbb{R}$ (not in the linear ...
1
vote
1answer
33 views

Is it possible to extract any encoded $x, y \in \mathbb{N^*}$ from $z=ax + by$

Is there any specific $a, b \in \mathbb{R}$, $\forall x,y \in \mathbb{N^*}$, take $z=a\cdot{}x+b\cdot{}y$ (then $z\in\mathbb{R}$), we can always extract $a,b$ from $z$. Here below are some trials I ...
1
vote
2answers
71 views

Maximal domain for composite functions.

Question $\mathbf 5$ If $f:(-\infty,1)\to R$, $f(x)=2\log_{\,e}(1-x)$ and $g:[-1,\infty)\to R$, $g(x)=3\sqrt{x+1}$, then the maximal domain of the function $f+g$ is ...
3
votes
1answer
64 views

Show that T is a linear transformation and find a, b, c

I'm having trouble understanding this question and the proper way to solve it. I don't understand the solution given and why this was the right way to answer it. Problem: For the vector space ...
0
votes
2answers
49 views

Show that the functions are linearly independent

Given the following functions: $f_1(x) = e^x$, $f_2(x)=sin(x)$, $f_3(x) =cos(x)$, $f_4(x) =x$, $f_5(x) =1$ I have to show that they build a basis of a subspace of a space of all functions ...
4
votes
1answer
1k views

“Well defined” function - What does it mean?

What does it mean for a function to be well-defined? I encountered with this term in an excersice asking to check if a linear transformation is well-defined.
1
vote
2answers
94 views

Injective and surjective functions on a matrix

Suppose we have a function $G:M_2(\mathbb R) \to S_2(\mathbb R)$ where $S_2(\mathbb R)$ is a symmetric matrix such that $ S_2(\mathbb R) = \left\{A = \begin{bmatrix} a & b\\ c & ...
0
votes
2answers
29 views

When is the species a going to equal species b?

Species $A = 2000e^{0.05t}$ Species $B = 5000e^{0.02t}$ When is species $A$ going to equal Species $B$?
0
votes
0answers
33 views

Expressing matrix as its orthogonal

This relates with my question in Proving sum of two matrices to be identity. Given $m<n$ and non singular $n\times n$ symmetric matrix $A$. Suppose that $H$ and $K$ be $m \times n$ and $n\times ...
1
vote
1answer
47 views

Diagnolization of a non-invertible matrix?

Let $A$ be a $n \times n$ matrix. Let $\mathcal{B}$ be a basis for the subspace formed by the columns of $A$. Can there exist a diagonal matrix $C$ such that: $$Cx_{\mathcal{B}} = ...
3
votes
3answers
91 views

Do these matrices exist?

Say you have some non-zero vector in $x$. Can you have two matrices $A$ and $B$ such that: $$Ax = Bx$$ if $A$ and $B$ aren't the identity matrix? This isn't homework, just curiosity.
2
votes
1answer
118 views

Operators between normed linear spaces

Prove or give a counterexample about an operator $T$ from a normed linear space $X$ to another normed linear space $Y$ ($T\colon X\to Y$) which is an additive operator (holds addition homomorphism), ...
0
votes
1answer
32 views

System of Equations- using profits

Question: Keller industries' profits were up $20,000 this year over last year. This was an increase of 25%. a. Let T represent the profit this year and L the profit from last year and write a ...
0
votes
1answer
42 views

group homomorphism?

Given: $ \varphi : ( S_4, \circ) ~~~\rightarrow ~~~(\Bbb{Z}/4\Bbb{Z}),$ $~~~~~~~~~$ $\sigma \longmapsto [\sigma(1)]$ Is this function a group homomorphism? My idea is, to calculate some values, but ...
1
vote
1answer
29 views

real values for a function

We know that $f(x)=ax^2+bx+c=0$ has two real solutions when $b^2-4ac \geq 0$ My question is, if we have a function \begin{equation} f(x)=\frac {D\ln((-0.5\sec x-1)/k)}{\ln a}-\frac ...
0
votes
1answer
22 views

Uniqueness of Function (inner product?)

Consider a bilinear function $f : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ that is symmetric, i.e. $$ f(A_1, A_2) = f(A_2, A_1) \quad\forall A_1, A_2 \in \mathbb{R}^n $$ and satisfies $$ ...
0
votes
1answer
71 views

What is the rank of the differentiation operator on Pn? What is the kernel?

For this question I was thinking of saying Pn(x)=Ax^n+Ax^(n-1)+...+Ax^2+Ax^1+1 and finding the first derivative P'n(x)=n.Ax^(n-1)+(n-1)Ax^(n-2)+...+2Ax^1+A+0 so in matrix form would get a n by 1 ...
1
vote
1answer
69 views

What values make this true $f(f(x)) = f(f(f(x)))$ but this not true $f(x) = f(f(x))$

Let $(V,K)$ be a finite vector space, If $f$ is a member of a $\mathrm L(V,V)$ where $V=\mathbb R^2$ and $K=\mathbb R$, what values for $x$ make $f(f(x)) = f(f(f(x)))$ true and $f(x) = f(f(x))$ not ...
0
votes
1answer
33 views

What does left composition mean in this question?

Consider the vector space of all linear transformations $L(V,V)$ on the vector space $(V,K)$ and a linear map $F:L(V,V)\to L(V,V)$ such that $F(a)= b \circ a$ for all $a\in L(V,V)$, where $b\in ...
0
votes
1answer
173 views

If $(V,k)$ is a finite-dimensional vector space, then the space of all linear transformations on $V$ is finite dim and find its dim?

My issue with this is the only way I know how to prove it is to set $\dim V=n$, but then that wouldn't make sense because the second part is find the $\dim$. What I was thinking is using the ...
2
votes
2answers
212 views

Find the matrix A of the linear transformation T(M)

I know that if I substitute the first matrix for $T(M)$ I see what T does to each of the basis vectors. I don't understand how that creates a $3\times 3$ matrix though. I was looking at this ...
2
votes
0answers
40 views

Bijection from the plane to itself that takes a circle to a circle must take a straight line to a straight line

Prove, that a bijection from the plane to itself that takes a circle to a circle must take a straight line to a straight line. There exists an elementary proof? I know this question can be found here ...
0
votes
1answer
56 views

Computing the limit.

Studying for a midterm. Compute the following limit: $$\lim_{x\to 4} \frac{x+4}{x^2+3x-4}$$ Factor the denominator: $$\lim_{x\to 4} \frac{x+4}{(x+4)(x-1)}$$ The $(x+4)s$ cancel out: $$\lim_{x\to ...
1
vote
2answers
39 views

Finding the limit $\lim_{x\to-\infty} (2x)/(2x-1)^2$.

Studying for a midterm: Let $f(x)=\frac{2x}{(2x-1)^2}$ Then $\lim_{x\to-\infty} f(x)$ is: Now keep in mind I'm shaky on how to do infinity limits. I have $f(x)=\frac{2x}{(2x-1)^2}$ Remove x by ...
0
votes
2answers
55 views

A function of $f\circ g$

This is studying for my midterm. Let $f(x)=x^2/(x+1)$ and $g(x)=2x-3$ A function of $f\circ g$ is: So I begin with the equation: $$x^2/(x+1) \cdot 2x-3$$ Add one to the denominator of the second ...
2
votes
3answers
62 views

Equation of a line parallel to $5x-3y=7$ That goes through the point (3,-1)

This is a study question in preparation for my midterm. It's multiple choice. The answers are: A) $y=(5/3)x-(7/3)$ B) $y=(3/5)x-(14/5)$ C) $y=(5/3)x-6$ D) $y=-(3/5)x+(4/5)$ Here is my process: ...
1
vote
2answers
73 views

Why are $R^n$ treated as $R^{n+1}$ spaces in $R^{n+1}$?

$y = x$ is a line in $R^2$ space. But if you graph $z = x$ in $R^3$ space, it's a plane: Both functions have the same relations, so why is one a plane but the other a line?
4
votes
0answers
38 views

Decomposability in the tensor product sense of functions of two variables

Let $S$ and $T$ be "nice" metric spaces, e.g. complete normed fields like $\Bbb R$, $\Bbb C$ or $\Bbb Q_p$. Let $F$ be a function $$ F:S\times T\longrightarrow K $$ where $K$ is a topological field ...
0
votes
0answers
49 views

Are rotations linear mappings?

Given a rotation matrix $\rho:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ with $det(\rho)=1$ and $c \in \mathbb{R}$ is it correct to say $\rho(cx)=c\rho(x)$?
1
vote
1answer
210 views

The surjectivity of the canonical projection map

Let $X$ be a subspace of a vector space $Y$. Define the canonical projection map $$\pi: Y \to Y/X$$ by $$\pi(y) \mapsto \{y\} + \mid X \mid.$$ So if $S_0$ is a subspace of $S$, then for any coset in ...
0
votes
1answer
47 views

Explain me context of these functions.

If the weight of a pet rabbit in pounds is a function of its age in years. Call this function g and let a be the the current age of the rabbit. Also, let h be the inverse of g. $g(a)+1$ $g(2a)$ ...
0
votes
0answers
30 views

What can I say about these two points… [duplicate]

Two points on the graph of $y=kx^p$ are labeled $A$ and $C$. Point $A$ has coordinates $(a,b)$, where $0<a<1$ and point $C$ has coordinates $(c,d)$, where $1<c$. If we are told that that ...