3
votes
2answers
63 views

Intuitive and convincing argument that functions are vectors

Back to school time again. As I'm discussing all the mathy stuff and insights gained over the summer, I cannot help but notice that many of my peers in second or third year undergrad cannot bridge the ...
0
votes
1answer
22 views

Blend n number of values by distance

I have n number of values which each have a distance that determens how much of the amount that should be blended. I've tried to illustrate my problem visually: The blue numbers is the values, the ...
1
vote
2answers
35 views

linear function that satisfies both conditions

I am having problems understanding how to solve this question. Find a linear function that satisfies both of the given conditions. $f(-1) = 5, f(1) = 6$ Thanks, Note: i have the answer, just need ...
1
vote
1answer
23 views

Multidimensional fitting of two data sets

My problem is the following: A laser gives out a bunch of data points which are reflected off a metal surface and recorded by a camera attached to the side of the laser. The image the camera receives ...
0
votes
0answers
23 views

What happens when scaling a rectangle using a pivot point?

With multitouch screens, you can pinch to zoom. When such a gesture is triggered you are supplied with: An x scale factor; A y scale factor; A x pivot point; A y pivot point. When I have a ...
0
votes
4answers
35 views

Surjective function - proving

$f: \mathbb{R}\to \mathbb{R}$ $f(x) = x^3 -2x^4$ In order to prove that $f$ is not surjective, my teacher told me to find that in most the $f$ is negative. And indeed, only for $0<x<0.5$ it's ...
1
vote
1answer
31 views

Surjective functions and cal'

$f,g: \mathbb{R}\to \mathbb{R}$ Both are also surjective functions. My question is if $f+g$ will be also surjective. I need to dis/prove it if it's true or false. Now, my friend told me it's false ...
0
votes
0answers
18 views

Is the following subset a vector subspace of F(R,R). How to prove that it is a subspace.

Is the following subset a vector subspace of F(R, R)? The set of functions f : R → R such that f(x + 2) = f(x) for all x ∈ R I know this is obviously a subspace, but the mark scheme didn't ...
1
vote
2answers
59 views

Is a norm on $R^n$ linear?

I was reading the book Linear Algebra Done Right by Axler. In the chapter on inner product space (Ch.6), he defines the norm of x on $R^n$ space as: $||x|| = \sqrt{x_1^2 + ... + x_n^2}$ and says: ...
0
votes
1answer
20 views

Graph exponential function

I am having problems understanding why $xe^x + 10e^x$ has two $(x,y)$ intercepts. I understand why there is one $(0,10)$, but am unclear on how to return $(-10,0)$. Any help would be much ...
3
votes
1answer
81 views

Is $f(t)=(\cos(t),\sin(t))$ a function?

In the Linear Algebra book we're using (Linear Algebra with Applications, Bretscher, p.129), the author defines this as the function of the unit circle. I understand why the equation of a circle ...
1
vote
3answers
31 views

Find that the given linear transform is a isomorphism

I'm studying Linear Algebra and I'm having trouble demonstrating that a function is a isomorphism, that is: "Given the linear transform $T: V \rightarrow W$, $T$ is a isomorphism if and only if it is ...
-1
votes
1answer
29 views

Understanding a definition for vector-spaces

Let $V$, a finite dimensional vector space, and $L$, a subspace of $V$. Let $T:V^*\rightarrow L^*$ defined as: $T(\varphi)(x)=\varphi(x)$ for all $\varphi \in V^*$. Prove $T$ is onto. Well, I'm ...
1
vote
1answer
23 views

$\ker S$ is not contained in $\ker T$ implies $\dim \Im T \ge 1$

Let $T,S:V\rightarrow W$.where $V$ is a finite vector space above $F$ and $W$ is one-dimensional vector-space above $F$ ($\dim W = 1$). It is given that $\ker S$ isn't contained in $\ker T$. Why is ...
1
vote
2answers
60 views

Having trouble understanding what ker f is

I am trying to understand what exactly 'ker f' is. My guess is that it is a set of all of the elements that were lost in the process of a mapping. For example: $f:A\to B$ $f = \left\{ \begin{align} ...
0
votes
0answers
49 views

Scale invariance property of function

Consider a function $f_j$:$\mathbb{F}_p$ $\longrightarrow$ $\mathbb{F}_p$ where the set $\mathbb{F}_p$ is defined as $\{0,1,2,\dots,p-1\}$. Clearly there are $p^p$ possible maps. Here the index $j$ ...
0
votes
1answer
25 views

Difference between horizontal and vertical line tests.

Trying to understand what the differnce is between a vertical and horizontal line test. If an equation fails the vertical line test, what does that tell you about the graph? If an equation fails the ...
0
votes
0answers
51 views

Finding criteria for a household financial budget falsification

I’m working on a financial problem about budget of households. Households in a state fill a form about their net budget in every year and our insurance company investigate their financial status and ...
2
votes
2answers
68 views

Constructing a function similar to x^3 between [0,1]

I'm trying to construct a function $f$, in order to normalize a dataset(obviously where all the element come from $[0,1] \in \mathbb{R}$. The big picture is that the envisioned $f: [0,1] \rightarrow ...
0
votes
1answer
34 views

Linear independence of a set of 'simple' functions

I have some $\alpha = \{f_0(x),f_1(x),\dots,f_n(x)\}$ Where having the $k$ in $f_k(x)$ equal $x$, meaning $k=x$ makes $f_k(x)=1$ or zero if $k \ne x$. Now I want to show $\alpha$ is linearly ...
7
votes
1answer
147 views

Determine an explicit expression for $f$.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function, bounded such that the space $\mathrm{lin}\{f_k(x)=f(x+k)∣k ∈\mathbb{N}\}$ is finite-dimensional. Determine an explicit ...
0
votes
0answers
30 views

Elementwise operations vs. matrix functions

Is there any notable connection in general case between elementwise matrix operations (such as matrix addition, scalar multiplication, Hadamard-product), and matrix functions (such as power of ...
1
vote
1answer
29 views

Kalman's controllability condition implies that the r.h.s. is “non-degenerate.”

Assume $f:\mathbb R^n\times\mathbb R\to\mathbb R^n$ is given by $f(x,u)=Ax+bu$ for some choice of coordinates $(x,u):x\in\mathbb R^n,u\in\mathbb R$, some constant matrix $A\in M_n(\mathbb R)$, and a ...
0
votes
1answer
38 views

Function Decomposition

How do I decompose a function when I'm given $f(g(x))$ and $f(x)$ and the required is $g(x)$? I done some searching on Google and most sites demonstrate the solution where it's left open, they just ...
0
votes
2answers
25 views

Construct a basis for a finite subspace of the Function Space V

Let X a non empty set, $F$ a field and $V$ the set of function of $X$ on $F$. If $f,g \in V$ and $\lambda \in F$ , $f +g ,\lambda f \in V$ are the functions such that $\forall x \in X$, ...
2
votes
2answers
68 views

Determining Whether a Function is Odd or Even from its Equation

I don't understand how this "Equation" is even being solved. I understand I have to substitute -x in for x but after that I don't know understand what's going on here. Please someone explain to me ...
1
vote
2answers
27 views

Inverse function (basic algbra math)

Consider the following function: $f(x) = {1 / (x-6) }$ Find a formula for the inverse of the function. Here is what have so far? $y = 1/(x-6)$ ---> $ x = 1/(y-6) $ But my embarrassing problem is ...
1
vote
1answer
33 views

How to write this function?

I do not want the answer given to me, I just want assistance. Problem: Marcus invests $750 in an account that pays 9.8% interest compounded annually. Write a function that describes the account ...
1
vote
0answers
28 views

Formula for calculation score based on distance

I try to write function for calculating scoring from distance in my game. I found something similar : link But I need the distance(x) to be between 0-31855000 meters and score between 0 to 1000. ...
1
vote
0answers
33 views

Is this functional linear?

I know it's trivial, but is this functional not linear? $\phi:\mathbb{R}[X]\ni p \rightarrow p(0)p \in \mathbb{R}[X]$ $$\phi(p+q)=(p+q)(0)\cdot(p+q)=(p(0)+q(0))\cdot(p+q)\ne\phi(p)+\phi(q)$$
1
vote
0answers
39 views

How to approach sketching sine and cosine graphs with transformations

Any tips or suggestions in sketching these graphs quickly, and in ONE go? In exams, I don't want to spend ages re-drawing the original sine/cosine graph, one by one, following each new ...
-1
votes
1answer
54 views

Let f : X → Y and g : Y → Z be bijective mappings. Show that gf is bijective

Let f : X → Y and g : Y → Z be bijective mappings. Show that gf, the composition of f and g, is bijective. I have that since f(x)=y, and g(y)=z we get g(f(x))=g(y)=z is this enough to show gf is ...
0
votes
1answer
56 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
91 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 ...
0
votes
1answer
56 views

Functions determined by characters are linearly independent?

Let $X$ be a set with an action on $\mathbb{Z}/N\mathbb{Z}$. For a Dirichlet character $\chi \pmod N$ we set $$R(\chi)=\left\{ f:X \to \mathbb{C} ~\mid~ f(l s)=\chi(l)f(s) \text{ for all } l\in ...
1
vote
2answers
35 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
38 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
36 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
44 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?
2
votes
0answers
58 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
30 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
85 views

Find basis of the annihilator set

$V$ $= \text{span}\{(1,2,3),(1,1,1)\}$ $\subseteq \mathbb{R}^3$. Find the vectors spanning $V^0$ in terms of the usual basis for $(\mathbb{R}^3)^*$. So we want linear functionals $f \in V^*$ such ...
0
votes
0answers
55 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
74 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
71 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
23 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
75 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
25 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
39 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
125 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 ...