how to find null space basis directly by matrix calculation
The problem of finding the basis for the null space of an $m \times n$ matrix $A$ is a well-known problem of linear algebra. We solve $Ax=0$ by Gaussian e...
Apr 14, 2026 1 VIEWS
Topic: General
5104 Articles 
Notation in the derivative of the hinge loss function
The hinge loss function (summed over $m$ examples): $$ l(w)= \sum_{i=1}^{m} \max\{0 ,1-y_i(w^{\top} \cdot x_i)\} $$ My calculation of the subgradient for ...
Apr 14, 2026 1 VIEWS
Is the universal quantification symbol $\forall\;$ known as "for any x" or "for all x" in First Order Logic & why is this different to Discrete Math?
I'm reading a book by Mark Tarver called Logic, Proof and Computation. Chapter 8 (starting p71) is about First Order Logic. On page 77 (of Chapter 8)...
Apr 14, 2026 1 VIEWS
Proof of Pasting Lemma
I am trying to understand the proof of the Pasting Lemma. I have found several proofs but I am missing something from all of them. Wikipedia has: Statemen...
Apr 14, 2026 1 VIEWS
Number of distinct path in a graph with $n$ vertices
Let $T = (V , E)$ be a tree with $|V | = n\geqslant 2$. How many distinct paths are there (as sub graphs) in $T$? I already have the answer to this questi...
Apr 14, 2026 1 VIEWS
How do pocket calculators calculate exponents?
I'd like to know specifically how a pocket calculator (TI calculators also apply) calculates $e^{0.1}$, and what methods or algorithms pocket calcula...
Apr 14, 2026 1 VIEWS
Matrix notation $i$ $j$
Let $A = \begin{bmatrix} a_1 & a_2 & \cdots & a_n \\ \end{bmatrix}$ be a $n \times n$ matrix such that $a_i \cdot a_i = 1$ for all $i$ and $a_...
Apr 14, 2026 1 VIEWS
How to answer matrices problems involving "meet" and "join"?
Problem: Given the matrix below find the meet and join of A and B. \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bma...
Apr 14, 2026 1 VIEWS
Use the Intermediate Value Theorem to show that $\cos(x)=x^3$ has a solution.
I am not sure if I am fully understanding how to solve this, but I think that, since the since $g(x)=\cos(x)$ and $g(x)=x^3$ are continuous everywhere, th...
Apr 14, 2026 1 VIEWS
Real coordinate, Cartesian, and Euclidean space
What is the modern way of defining these concepts? (Real coordinate space, Cartesian coordinate system, Euclidean space). As far as I can tell: Real coord...
Apr 14, 2026 1 VIEWS
Ordinary Induction vs. Strong Induction (on the same example)
I'm looking for a solid explanation to grow in intuition for strong induction after having a decent foundation in ordinary induction. For example, Th...
Apr 14, 2026 1 VIEWS
When can't you change the order of integration?
Apparently there are integrals which you can express as $\int_A \int_B f(x, y) \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}y$ but not $\int_C \int_D f(x, ...
Apr 14, 2026 1 VIEWS