Improve Your Algebra Skills with Practice Problems

How do you solve and graph ${\displaystyle 5-(x+4)\le -3}$ ?

How do you solve ${\displaystyle 11\le y-26}$ ?

Using Quaternion Coefficients to transform a vector from one reference frame to another
I hope this question is not too trivial, and I welcome any pointers to good resources for this problem. I am not familiar with quaternions and have never had to use them before--all my learning about them has been my reading tonight. But most of the resources I've found haven't been too helpful, and I suspect I'm missing vocabulary that I might need to get to the information I need.
Suppose I have sensor which provides the quaternion coefficients for its relative rotation to some "primary" frame of reference. These coefficients are available in a vector, $⟨<!--\; ⟨\; -->w,x,y,z⟩<!--\; ⟩\; -->$, corresponding to the quaternion $w+xi+yj+zk$. I have a second vector, provided by the same sensor, which indicates acceleration, and is provided in the form $⟨<!--\; ⟨\; -->X,Y,Z⟩<!--\; ⟩\; -->$. This second vector provides the data from the sensor's frame of reference, not with respect to the "primary" frame of reference.
My goal is to transform the acceleration vector to the primary reference frame, so I can find the components of that vector with respect to my primary reference frame. This way I can determine the acceleration in each direction in the primary frame of reference.
From terms I've encountered, and a handful of other posts on similar issues, it seems that I would need to use some sort of rotation matrix, or possibly--since I have the quaternion already--use its inverse to return to the primary frame. But I am unsure of this, and wouldn't know how to do that without a better understanding of the methods involved.
I am hoping someone could point me to clear resources on this sort of problem, or explain the procedure (ideally with a simple example), and would be deeply grateful for any assistance.

How to interpret data from Mann-Whitney U Test
So I am doing a research project and I was told to do the Mann Whitney U Test. The research is examining male and female work experiences by them ranking statements from 1-6 (1 being strongly agree and 7 being strongly disagree). The goal is to see if their is a difference in work experience between the two groups. Using an online calculator these were my results: Key: 1st group/2nd group
Variable: Female/Male Observations: 720/936
Mean: 3.099/3.042 SD: 1.715/1.668
Mann-Whitney test / Two-tailed test:
U 341978.5
U (standardized) 0.532
Expected value 336960
Variance (U) 89023289.87
p-value (Two-tailed) 0.595
alpha 0.05
So I have all this great data. The first part makes sense to me but the data in the second part I don't know how to read/analyze. Can somebody explain. Thank you!

How to interpret a too small chi-square ${\mathrm{\chi <!--\; \chi \; -->}}^2$ value?
I use the Chi-square (i.e., ${\mathrm{\chi <!--\; \chi \; -->}}^2$) goodness-of-fit test to measure the distance between chunks of data and a theoretical distribution.
For most of the data being tested the results make sense. I got from time to time unusual results, which correspond to very small values (i.e., between 10 and 100). The sample size is not the problem, there is at least 5 elements for each symbol.
After a careful analysis of the observed data, the frequency of each symbol in the data being tested is very close to the theoretical distribution as the ${\mathrm{\chi <!--\; \chi \; -->}}^2$ value suggest.
But it seems to be too perfect for me. It is unlikely according to the chi-square distribution function that a given value would occur (i.e., $z_{0.0001}$).
Should I reject the null hypothesis because the data is too good to be true ?
I did not find clear explanations or I missed something.
How to interpret unlikely values in the left tail of the chi-square distribution ?

Partial Fractions Question:$\frac{1}{s(n-<!--\; -\; -->s)}$
How do you decompose the fraction to two fractions?
(s and n are variables).

How to compute Bias and Variance for the given scenarios?
I'm currently studying the "Learning from data" course - by Professor Yaser Abu, and I do not get the "bias-variance tradeoff" part of it. Actually, the concepts are fine − the math is the problem.
In the lecture 08, he defined bias and variance as follows:
$\text{Bias}={\mathbb{E}}_{\mathbf{x}}[(\stackrel{¯<!--\; ¯\; -->}{g}(\mathbf{x})-<!--\; -\; -->f(\mathbf{x}){)}^2]$, where $\stackrel{¯<!--\; ¯\; -->}{g}(\mathbf{x})={\mathbb{E}}_{\mathcal{D}}[g^{(\mathcal{D})}(\mathbf{x})]$
$\text{Var}={\mathbb{E}}_{\mathbf{x}}\left[{\mathbb{E}}_{\mathcal{D}}[(g^{(\mathcal{D})}(\mathbf{x})-<!--\; -\; -->\stackrel{¯<!--\; ¯\; -->}{g}(\mathbf{x}){)}^2]\right]$
To clarify the notation:
D means the data set $({\mathbf{x}}_1,y_1),\cdots <!--\; \cdots \; -->,({\mathbf{x}}_n,y_n)$.
g is the function that approximates $f$; i.e., I'm estimating $f$ by using g. In this case, g is chosen by an algorithm $\mathcal{A}$ in the hypothesis set $\mathcal{H}$.
After that, he proposed an example that was stated in the following manner:
Example: Let $f(x)=\sin <!--\; \; -->(\mathrm{\pi <!--\; \pi \; -->}x)$ and a data set $\mathcal{D}$ of size N=2. We sample x uniformly in [−1,1] to generate $({\mathbf{x}}_1,y_1)$ and $({\mathbf{x}}_2,y_2)$. Now, suppose that I have two models, ${\mathcal{H}}_0$ and ${\mathcal{H}}_1$.
${\mathcal{H}}_0:h(x)=b$
${\mathcal{H}}_1:h(x)=ax+b$
${\mathcal{H}}_0:h(x)=b$
${\mathcal{H}}_1:h(x)=ax+b$
For ${\mathcal{H}}_0$, let $b=\frac{y_1+y_2}{2}$. For ${\mathcal{H}}_1$, choose the line that passes through $({\mathbf{x}}_1,y_1)$ and $({\mathbf{x}}_2,y_2)$.
Simulating the process as described, he states that:
Looking for ${\mathcal{H}}_0$, $\text{Bias}\approx <!--\; \approx \; -->0.50$ and $\text{Var}\approx <!--\; \approx \; -->0.25$.
Looking for ${\mathcal{H}}_1$, $\text{Bias}\approx <!--\; \approx \; -->0.21$ and $\text{Var}\approx <!--\; \approx \; -->1.69$.
Here is my main question: How can one get these results analytically?
I've tried to solve the integrals (it didn't work) that came from the $\mathbb{E}[\cdot <!--\; \cdot \; -->]$, but I'm not sure if
I'm interpreting in the right way which distribution is which. For example, how to evaluate ${\mathbb{E}}_{\mathcal{D}}[g^{(\mathcal{D})}(\mathbf{x})]$ (it is the same as evaluating ${\mathbb{E}}_{\mathcal{D}}\left[b\right]$ or ${\mathbb{E}}_{\mathcal{D}}[ax+b]$, for ${\mathcal{H}}_0$ and ${\mathcal{H}}_1$, respectively, right?)? The random variable which has uniform distribution over [−1,1] is x, right? Thus
${\mathbb{E}}_{\mathbf{x}}[\cdot <!--\; \cdot \; -->]$ is evaluated with respect to a random variable that follows $U[-<!--\; -\; -->1,1]$] distribution, right?
If anyone could help me to understand at least one of the two scenarios, by achieving the provided numbers for the Bias and Var quantities; it would be extremely helpful.
Thanks in advance,
André

How exactly is this happening?
I was studying Derivative and my book says if:
$y=\sqrt{(1+x^2{)}^3}$
Then its derivative is:
$y^{\prime }=\frac{3(2x)(1+x^2{)}^2}{2\sqrt{(1+x^2{)}^{-<!--\; -\; -->1}}}=3x\sqrt{(1+x^2)}$
I can't understand how the writer has changed the first derivative fraction into the second one. In other words, how did he simplify?
Note: I'm really really basic, so please explain in details.
Thank you.

Say we have a sequence of random variables ($X_n$) with mean ${\mathrm{\mu <!--\; \mu \; -->}}_n$ and variance ${\mathrm{\sigma <!--\; \sigma \; -->}}_n^2$, that converges in probability towards a random variable $X$. Can we reason that mean and variance sequences are bounded? Do we need some other assumptions, for example, $E(|X|)<\mathrm{\infty <!--\; \infty \; -->}$, or perhaps if we assume that each $X_n$ is normal? Thank you in advance.

I have been working on the following problem during a Quantum Mechanics course I am taking as part of my postgraduate in Mathematics, and I feel I am missing something very simple.

Let $\mathrm{\xi <!--\; \xi \; -->}\in <!--\; \in \; -->{\mathbb{C}}^n$, $||\mathrm{\xi <!--\; \xi \; -->}||=1$ and $P_{\mathrm{\xi <!--\; \xi \; -->}}:{\mathbb{C}}^n\to <!--\; \to \; -->{\mathbb{C}}^n$ be the projector on $\mathrm{\xi <!--\; \xi \; -->}$. That is, $P_{\mathrm{\xi <!--\; \xi \; -->}}\mathrm{\psi <!--\; \psi \; -->}=(\mathrm{\psi <!--\; \psi \; -->},\mathrm{\xi <!--\; \xi \; -->})\mathrm{\xi <!--\; \xi \; -->}$ for any $\mathrm{\psi <!--\; \psi \; -->}\in <!--\; \in \; -->{\mathbb{C}}^n$.

1. Show that $P_{\mathrm{\xi <!--\; \xi \; -->}}$ is a density matrix.
2. Let $\mathrm{\omega <!--\; \omega \; -->}$ be the state described by $P_{\mathrm{\xi <!--\; \xi \; -->}}$. Show that the expectation $⟨<!--\; ⟨\; -->A{⟩<!--\; ⟩\; -->}_{\mathrm{\omega <!--\; \omega \; -->}}=(A\mathrm{\xi <!--\; \xi \; -->},\mathrm{\xi <!--\; \xi \; -->})$ for any $A\in <!--\; \in \; -->\mathcal{A}$, where $\mathcal{A}$ is the space of linear Hermitian operators on ${\mathbb{C}}^n$.
3. Let ${\mathrm{\lambda <!--\; \lambda \; -->}}_j$ be a simple eigenvalue of $A$ and ${\mathrm{\psi <!--\; \psi \; -->}}_j$, $||{\mathrm{\psi <!--\; \psi \; -->}}_j||=1$ be an eigenvector corresponding to ${\mathrm{\lambda <!--\; \lambda \; -->}}_j$. Suppose that the observable A is measured when the quantum system is in the state $\mathrm{\omega <!--\; \omega \; -->}$ described in part 2. Show that the probability that the outcome of the measurement coincides with ${\mathrm{\lambda <!--\; \lambda \; -->}}_j$ is equal to $|({\mathrm{\psi <!--\; \psi \; -->}}_j,\mathrm{\xi <!--\; \xi \; -->}){|}^2$.

I have managed to work my way through parts 1 and 2 but am unsure on how to proceed with part 3. Any hints would be greatly appreciated.

How do you solve ${\displaystyle -3t\ge 39}$ ?

How to do 24.89 / 26.15 with long division?
How to do 24.89 / 26.15 with long division? Question is to find applied tax if 24.89 is pre-tax and 26.15 is post-tax. I'm aware of how to solve this with a calculator (1-(24.89/26.15))= ~.0482 = ~ 4-5 cents

show that $\frac{c^2}{a}+\frac{a^2}{c}+16\sqrt{ac}\ge <!--\; \ge \; -->9(a+c)$
Let $a,c>0$ show that
${\displaystyle \frac{c^2}{a}}+{\displaystyle \frac{a^2}{c}}+16\sqrt{ac}\ge <!--\; \ge \; -->9(a+c)$
It seem AM-GM inequality.How?

How can I proceed?
For a arbitrary positive $\mathrm{\epsilon <!--\; \epsilon \; -->}$, is there a $n_{\mathrm{\epsilon <!--\; \epsilon \; -->}}\in <!--\; \in \; -->\mathbb{N}$ such that
$\underset{(a_n{)}_{n=1}^{\mathrm{\infty <!--\; \infty \; -->}}\in <!--\; \in \; -->X}{sup}\left({\displaystyle \underset{n\ge <!--\; \ge \; -->n_{\mathrm{\epsilon <!--\; \epsilon \; -->}}}{\sum <!--\; \sum \; -->}|a_n|}\right)\le <!--\; \le \; -->\mathrm{\epsilon <!--\; \epsilon \; -->},$
where $X$ is bounded in ${\mathrm{\ell <!--\; \ell \; -->}}_1$?
PS: ${\mathrm{\ell <!--\; \ell \; -->}}_1$ is a sequence space such that:
${\mathrm{\ell <!--\; \ell \; -->}}_n=\{(a_i{)}_{i=1}^{\mathrm{\infty <!--\; \infty \; -->}};a_i\in <!--\; \in \; -->\mathbb{K},\mathrm{\forall <!--\; \forall \; -->}i\in <!--\; \in \; -->\mathbb{N}\text{ and }{\displaystyle \underset{i=1}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}|a_i{|}^n<\mathrm{\infty <!--\; \infty \; -->}}\},$
where $\mathbb{K}$ is the scalar space. Also, the norm $||\cdot <!--\; \cdot \; -->|{|}_n$ is given by
$||(a_i{)}_{i=1}^{\mathrm{\infty <!--\; \infty \; -->}}|{|}_n={\left({\displaystyle \underset{i=1}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}|a_i{|}^n}\right)}^{\frac{1}{n}}.$
My attempt is:
For all $a\in <!--\; \in \; -->X$, there is a $M>0$ such that $||a|{|}_1\le <!--\; \le \; -->M$. Than take a sequence $(a_j{)}_{j=1}^{\mathrm{\infty <!--\; \infty \; -->}}$ in $X$ and we have
$\left({\displaystyle \underset{j=1}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}|a_j|}\right)=||(a_j{)}_{j=1}^{\mathrm{\infty <!--\; \infty \; -->}}|{|}_1\le <!--\; \le \; -->M.$

Soo any help.

How to solve advanced fraction problem with algebra
Good evening guys, ok so ive been stumped on this for some time and am a bit rusty with my fractions so i was wondering if i could get some help on answering this
I have to try and write this fraction on its own and in its most basic form (simple)
The problem:
${\displaystyle \frac{(\frac{a^3-<!--\; -\; -->3a^2}{8a^2-<!--\; -\; -->4a})}{(\frac{a^2-<!--\; -\; -->9}{2a^2+5a-<!--\; -\; -->3})}}$
Am really stuck, any help would be appreciated!

the sine function is an odd function, for a negative number $u,\sin <!--\; \; -->2u=-<!--\; -\; -->2\sin <!--\; \; -->u\cos <!--\; \; -->u$. Is it true or false and why?

Three Questions on Interpreting the Outcome of the Probability Density Function
I have three questions: What is the interpretation of the outcome of the Probability Density Function (PDF) at a particular point? How is this result related to probability? What we exactly do when we maximize the likelihood?
To better explain my questions:
(i) Consider a continuous random variable X with a normal distribution such that $\mathrm{\mu <!--\; \mu \; -->}=1.5$ and ${\mathrm{\sigma <!--\; \sigma \; -->}}^2=2$. If we evaluate the PDF at a particular point, say 3.4, using the formula:
$f(X)=\frac{1}{\sqrt{2\mathrm{\pi <!--\; \pi \; -->}{\mathrm{\sigma <!--\; \sigma \; -->}}^2}}{e}^{\frac{-<!--\; -\; -->{\left(X-<!--\; -\; -->\mathrm{\mu <!--\; \mu \; -->}\right)}^2}{2{\mathrm{\sigma <!--\; \sigma \; -->}}^2}},$
we get $f(3.4)=0.1144$. How we interpret this value?
(ii) I previously read that the result of 0.1144 is not necessarily the probability that X takes the value of 3.4. But how the result is related to probability concept?
(iii) Consider a sample of the continuous random variable X of size N=2.5, such that $X_1=2$ and $X_2=3.5$. We can use this sample to maximize the log-likelihood:
$max\ln <!--\; \; -->L(\mathrm{\mu <!--\; \mu \; -->},\mathrm{\sigma <!--\; \sigma \; -->}|X_1,X_2)=\ln <!--\; \; -->f(X_1)+\ln <!--\; \; -->f(X_2)$
If f(X) is not exactly a probability, what are we maximizing? Some texts detail that "we are maximizing the probability that a model (set of parameters) reproduces the original data". Is this phrase incorrect?

I am trying to understand the Borel-Cantelli lemma, however, I cannot understand what exactly is "infinitely often". Can someone please explain "infinitely often" by a simple example?

For a measure space $(X,\mathcal{E},\mathrm{\mu <!--\; \mu \; -->})$ we introduced the complete measure space ${\mathcal{E}}_{\mathrm{\mu <!--\; \mu \; -->}}$ defined as:
${\mathcal{E}}_{\mathrm{\mu <!--\; \mu \; -->}}:=\{A\in <!--\; \in \; -->\mathcal{P}(X)|\mathrm{\exists <!--\; \exists \; -->}B,C\in <!--\; \in \; -->\mathcal{E}\text{ with }A\mathrm{△<!--\; △\; -->}B\subset <!--\; \subset \; -->C\text{ and }\mathrm{\mu <!--\; \mu \; -->}(C)=0\}$
and we define the corresponding complete measure as
${\stackrel{¯<!--\; ¯\; -->}{\mathrm{\mu <!--\; \mu \; -->}}}_1(A)=\mathrm{\mu <!--\; \mu \; -->}(B)$
On the internet I've found exclusively an alternative definition, for which we define
$\mathcal{N}:=\{N\subset <!--\; \subset \; -->X|\mathrm{\exists <!--\; \exists \; -->}M\in <!--\; \in \; -->\mathcal{E}\text{ mit }\mathrm{\mu <!--\; \mu \; -->}(M)=0\text{ und }N\subset <!--\; \subset \; -->M\}$
and then the complete measure, I'll name it $\stackrel{¯<!--\; ¯\; -->}{\mathcal{E}}$ is:
$\stackrel{¯<!--\; ¯\; -->}{\mathcal{E}}:=\{D\cup <!--\; \cup \; -->N|D\in <!--\; \in \; -->\mathcal{E},N\in <!--\; \in \; -->\mathcal{N}\}$
and a measure
${\stackrel{¯<!--\; ¯\; -->}{\mathrm{\mu <!--\; \mu \; -->}}}_2(D\cup <!--\; \cup \; -->N)=\mathrm{\mu <!--\; \mu \; -->}(D)$
I find it much easier to work with the alternative definition from the internet, and I was wondering what advantages the one from my professor would have. And secondly, if both are complete measure spaces with respect to $\mathcal{E}$, is also ${\mathcal{E}}_{\mathrm{\mu <!--\; \mu \; -->}}=\stackrel{¯<!--\; ¯\; -->}{\mathcal{E}}$ and ${\mathrm{\mu <!--\; \mu \; -->}}_1={\mathrm{\mu <!--\; \mu \; -->}}_2$? How could I start proving it?

Meaning of the word "axiom"
One usually describes an axiom to be a proposition regarded as self-evidently true without proof.
Thus, axioms are propositions we assume to be true and we use them in an axiomatic theory as premises to infer conclusions, which are called "theorems" of this theory.
For example, we can use the Peano axioms to prove theorems of arithmetic.
This is one meaning of the word "axiom". But I recognized that the word "axiom" is also used in quite different contexts.
For example, a group is defined to be an algebraic structure consisting of a set G, an operation $G\times <!--\; \times \; -->G\to <!--\; \to \; -->G:(a,b)\mapsto <!--\; \mapsto \; -->ab$, an element $1\in <!--\; \in \; -->G$ and a mapping $G\to <!--\; \to \; -->G:a\mapsto <!--\; \mapsto \; -->a^{-<!--\; -\; -->1}$ such that the following conditions, the so-called group axioms, are satisfied:
$\mathrm{\forall <!--\; \forall \; -->}a,b,c\in <!--\; \in \; -->G.(ab)c=a(bc)$
$\mathrm{\forall <!--\; \forall \; -->}a\in <!--\; \in \; -->G.1a=a=a1$ and
$\mathrm{\forall <!--\; \forall \; -->}a\in <!--\; \in \; -->G.a{a}^{-<!--\; -\; -->1}=1=a^{-<!--\; -\; -->1}a$
Why are these conditions (that an algebraic structure has to satisfy to be called a group) called axioms? What have these conditions to do with the word "axiom" in the sense specified above? I am really asking about this modern use of the word "axiom" in mathematical jargon. It would be very interesting to see how the modern use of the word "axiom" historically developed from the original meaning.
Now, let me give more details why it appears to me that the word is being used in two different meanings:
As peter.petrov did, one can argue that group theory is about the conclusions one can draw from the group axioms just as arithmetic is about the conclusions one can draw from the Peano axioms. But in my opinion there is a big difference: while arithmetic is really about natural numbers, the successor operation, addition, multiplication and the "less than" relation, group theory is not just about group elements, the group operation, the identity element and the inverse function. Group theory is rather about models of the group axioms. Thus: The axioms of group theory are not the group axioms, the axioms of group theory are the axioms of set theory.
Theorems of arithmetic can be formalized as sentences over the signature (a. k. a. language) $\{0,s,+,\cdot <!--\; \cdot \; -->\}$, while theorems of group theory cannot always be formalized as sentences over the signature $\{\cdot <!--\; \cdot \; -->,1{,}^{-<!--\; -\; -->1}\}$. Let me give an example: A typical theorem of arithmetic is the case n=4 of Fermat's last theorem. It can be formalized as follows over the signature $\{0,s,+,\cdot <!--\; \cdot \; -->\}$:
$\mathrm{\neg <!--\; \neg \; -->}\mathrm{\exists <!--\; \exists \; -->}x\mathrm{\exists <!--\; \exists \; -->}y\mathrm{\exists <!--\; \exists \; -->}z(x\ne 0\wedge <!--\; \wedge \; -->y\ne 0\wedge <!--\; \wedge \; -->z\ne 0\wedge <!--\; \wedge \; -->x\cdot <!--\; \cdot \; -->x\cdot <!--\; \cdot \; -->x\cdot <!--\; \cdot \; -->x+y\cdot <!--\; \cdot \; -->y\cdot <!--\; \cdot \; -->y\cdot <!--\; \cdot \; -->y=z\cdot <!--\; \cdot \; -->z\cdot <!--\; \cdot \; -->z\cdot <!--\; \cdot \; -->z).$
A typical theorem of group theory is Lagrange's theorem which states that for any finite group G, the order of every subgroup H of G divides the order of G. I think that one cannot formalize this theorem as a sentence over the "group theoretic" signature $\{\cdot <!--\; \cdot \; -->,1{,}^{-<!--\; -\; -->1}\}$; or can one?