# Tagged Questions

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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $\ker S$ 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 ...
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### Linear Equation Formulas for specific questions

Im trying to figure out how to do this problem, but it is just extremely confusing to understand how too do. A cricket chirps at different retes depending on temperature. You can estimate the ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\forall x \in X$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 $e_2 = \{ (1, 0),(2, 1) \}$. ...
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### 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 ...
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### 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 ...
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 ...