Demystify College Algebra with Step-by-Step Examples and Expert Guidance

See above. I am trying to re-teach myself mathematics in a different manner than is formally taught (i.e., set theory, number theory, mathematical logic, abstract algebra, discrete math and then precalculus (college algebra and analytical geometry). These are the pillars to upper level mathematics correct?

I just started my first upper level undergrad course, and as we were being taught vector spaces over fields we quickly went over fields. What confused me that the the set {0, 1, 2} was a field. However, to my understanding, that set doesn't satisfy the axiom, "For every element a in F, there is an element b such that a+b=0", among others. Can someone help clarify where my understanding is off.
Also one my friend states "for every prime power p^n, there exists a field with $p^n$ elements" and then doesn't expand on it. If someone could give an example or proof i would be very grateful.

I took abstract algebra I (focused on group theory) last semester before exposure to linear algebra, and am now starting a linear algebra course. I greatly enjoyed abstract algebra, and I'm curious which concepts will transfer over to linear algebra; any perspectives or connections that could aid in understanding would be helpful.

I'm in highschool and just started integration. I found the following problem in an old question paper and I find it very challenging.
${\int <!--\; \int \; -->}_0^1\sqrt{x^2-<!--\; -\; -->2x+1}dx$
so I simplified it algebraically to
${\int <!--\; \int \; -->}_0^1\sqrt{(x-<!--\; -\; -->1{)}^2}dx$
which of course is
${\int <!--\; \int \; -->}_0^1|x-<!--\; -\; -->1|dx$
as the absolute value is a linear function over $x\in <!--\; \in \; -->[0,\mathrm{\infty <!--\; \infty \; -->})$ so I proceed to evaluate it as $x^2/2-<!--\; -\; -->x$ for upper limit 1 and lower limit 0 which is $((1{)}^2/2-<!--\; -\; -->1)-<!--\; -\; -->(0-<!--\; -\; -->0)$ and equals $-<!--\; -\; -->\frac{1}{2}$, but according to wolfram alpha it is $\frac{1}{2}$.
Please explain at beginner level.

What is the Z-score for a 10% confidence level (i.e. 0.1 pvalue)?
I want the standard answer used for including in my thesis write up. I googled and used excel to calculate as well but they are all slightly different.
Thanks.

I was going through a simple proof written for the existence of supremum. When I tried to write a small example for the argument used in the proof, I got stuck. The proof is presented in Vector Calculus, Linear Algebra, And Differential Forms written by Hubbard. Here is the theorem.
Theorem: Every non-empty subset $X\subset <!--\; \subset \; -->\mathbb{R}$ that has an upper bound has a least upper bound $(supX)$.
Proof: Suppose we have $x\in <!--\; \in \; -->X$ and $x\ge <!--\; \ge \; -->0$. Also, suppose $\mathrm{\alpha <!--\; \alpha \; -->}$ is a given upper bound.
If $x\ne <!--\; \ne \; -->\mathrm{\alpha <!--\; \alpha \; -->}$, then there is a first $j$ such that $j^{th}$ digit of $x$ is smaller than the $j^{th}$ of $\mathrm{\alpha <!--\; \alpha \; -->}$. Consider all the numbers in $[x,a]$ that can be written using only $j$ digits after the decimal, then all zeros. Let $b_j$ bt the largest which is not an upper bound. Now, consider the set of numbers $[b_j,a]$ that have only $j+1$ digits after the decimal points, then all zeros.
The proof continues until getting $b$ which is not an upper bound.
Now, let's assume that $X=\{0.5,2,2.5,\cdots <!--\; \cdots \; -->,3.23\}$ where 3.23 is the largest value in the set and $\mathrm{\alpha <!--\; \alpha \; -->}=6.2$. I select 2 as my $x$. Then, the value of $j$ is one. Hence, the set is defined as [2.1,2.2.,⋯,2.9,6.2]. In this case, $b_j$ is 2.9.
If create a new set based on $b_j$, doesn't that become [2.91,2.92.,⋯,2.99,6.2]. If so, $b_{j+1}=2.99$ and I'd just add extra digits behind this number. In other words, I'd never go to the next level. I am probably misinterpreting the proof statement.
Two important details after receiving some comments are as follows.
1. "By definition, the set of real numbers is the set of infinite decimals: expressions like 2.95765392045756..., preceded by a plus or a minus sign (in practice the plus sign is usually omitted). The number that you usually think of as 3 is the infinite decimal 3.0000... , ending in all zeroes."
2. "The least upper bound property of the reals is often taken as an axiom; indeed, it characterizes the real numbers, and it sits at the foundation of every theorem in calculus. However, at least with the description above of the reals, it is a theorem, not an axiom."

Qualitatively (or mathematically "light"), could someone describe the difference between a matrix and a tensor? I have only seen them used in the context of an undergraduate, upper level classical mechanics course, and within that context, I never understood the need to distinguish between matrices and tensors. They seemed like identical mathematical entities to me.
Just as an aside, my math background is roughly the one of a typical undergraduate physics major (minus the linear algebra).

I understand the process for how Eigenvalues are involved in Differential Equations. If you have Differential System of Equations like this
${\stackrel{\to <!--\; \to \; -->}{x}}^{\prime }=\left(\begin{array}{cc}2& 1\\ 0& 1\end{array}\right)\stackrel{\to <!--\; \to \; -->}{x}$
The solution to that System of Differential Equations is a Linear Combination of e to the power of the eigenvalues times the corresponding eigenvectors.
$\stackrel{\to <!--\; \to \; -->}{x}=C_1{e}^{2t}\left(\begin{array}{c}1\\ 0\end{array}\right)+C_2{e}^t\left(\begin{array}{c}-<!--\; -\; -->1\\ 1\end{array}\right)$
However, what I am struggling with figuring out how Generalized Eigenvalues translate to the solutions in Differential Equations. If you take this System of Differential Equations
${\stackrel{\to <!--\; \to \; -->}{x}}^{\prime }=\left(\begin{array}{ccc}1& 0& 0\\ 3& 1& 0\\ 6& 3& 1\end{array}\right)\stackrel{\to <!--\; \to \; -->}{x}$
The solution to this Differential Equations is
$\stackrel{\to <!--\; \to \; -->}{x}=C_1{e}^t\left(\begin{array}{c}1\\ 3t\\ 6t+\frac{9}{2}{t}^2\end{array}\right)+C_2{e}^t\left(\begin{array}{c}0\\ 1\\ 3t\end{array}\right)+C_3{e}^t\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)$
But the eigenvectors of this matrix (including the generalized eigenvectors) are
${\stackrel{\to <!--\; \to \; -->}{v}}_1=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right),{\stackrel{\to <!--\; \to \; -->}{w}}_1=\left(\begin{array}{c}0\\ 1\\ 3\end{array}\right),{\stackrel{\to <!--\; \to \; -->}{w}}_2=\left(\begin{array}{c}1\\ 1\\ 3\end{array}\right)$
These eigenvectors do share some similarities to the solution to the System of Equations. However, the
$6t+\frac{9}{2}{t}^2$
term in the solution is one I have no idea how generalized eigenvectors relate. May someone please explain how these eigenvectors translate into Differential Equations?

I've never fully understood the connection between Riemann surfaces and algebraic varieties. I'm particularly interested in the case of the modular curve of level N--I know how the Riemann surface is constructed by taking a quotient of the upper half-plane by the action of a congruence subgroup of the modular group, but not how the resulting manifold translates into a curve. From what I've read, it appears that the associated curve is defined by equations satisfied by functions defined on the manifold, but I don't understand which functions are involved in these equations. What exactly is the relation between the two types of objects?

We can do row operations without changing det(A)'
But let's say I have an arbitrary upper triangular matrix U
$U=\left[\begin{array}{ccc}a& a& a\\ 0& b& b\\ 0& 0& c\end{array}\right]$
And I perform the following row operations on U to bring it to U′
$\frac{1}{a}{R}_1\to <!--\; \to \; -->R_1$
$\frac{1}{b}{R}_2\to <!--\; \to \; -->R_2$
$\frac{1}{c}{R}_3\to <!--\; \to \; -->R_3$
Then U; is:
$U^{\prime }=\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]$
But now det(U)=abc and det(U′)=1, thus
$det(U)\ne <!--\; \ne \; -->det(U^{\prime })$
All I've done is perform row operations on U to bring it to U′, but by performing those row operations, their determinants lose equality. How can that be possible?
So how is this seeming contradiction is resolved. I'm assuming that I must have some misconception either on row operations or on determinants.
Furthermore on a deeper level, what geometric interpretation/meaning does scaling the rows as I've done bringing U to U′, have on the determinant? Since the determinants of U and U′ are obviously no longer equal, geometrically what is this scaling doing to the determinant?

Given $S{L}_2(\mathbb{Z})$, what interpretation is available for the hyperbolic elements? What is true, if we consider a congruence subgroup?
I heard that there is a connection with certain class numbers.