"Elements of Abstract and Linear Algebra" by Edwin H. Connell is a freely available university-level textbook that provides a rigorous introduction to both abstract and linear algebra.
The book covers the foundations of mathematics, sets, relations, functions, mathematical induction, group theory, rings, fields, matrices, determinants, vector spaces, linear transformations, homomorphisms, eigenvalues and eigenvectors, characteristic polynomials, diagonalization, Jordan canonical forms, inner product spaces, and the Gram–Schmidt orthogonalization process.
I recommend adding this one to your personal library, both for studying these subjects and as a reference for reviewing their fundamental concepts.
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"Contemporary Mathematics" is a free textbook that introduces fundamental mathematical concepts over nearly 1,900 pages through clear explanations, numerous worked examples and exercises.
The book covers set theory, logic, Venn diagrams, mathematical statements and proofs, De Morgan’s laws, number systems, elementary number theory, prime numbers, rational and irrational numbers, arithmetic sequences, geometric sequences, numeral systems, calculations in different bases, and many other topics, providing a broad overview of foundational mathematics.
It is particularly useful for high school students, and I recommend keeping it if you have children studying mathematics at that level. The book is well structured, accessible, and enjoyable to browse, making it a valuable resource for both learning and reviewing mathematical concepts.
https://t.co/QK3RHNInkq
How do we know dark matter exists if we cannot see it?
Galaxies rotate faster than their visible matter can explain, and their gravity bends light from distant galaxies. Both reveal mass we cannot account for with stars and gas alone. We can measure its gravitational effects, but after decades of research, we still do not know what dark matter actually is.
An electron in a hydrogen atom has a characteristic speed of about 2,190 km/s in its ground state, roughly 0.73% the speed of light.
Niels Bohr’s 1913 model treated the electron as moving around the proton in a fixed orbit. Modern quantum mechanics tells a different story: an electron’s orbital describes where it is likely to be found, not a definite path through space.
The surprising part is that we can calculate its speed without being able to trace its journey like a tiny planet.
A single number is already a tensor.
Add one index and it becomes a vector. Add a second and you have a matrix. Add a third and the array becomes a cube of numbers. Each step simply increases the number of indices needed to pick out one entry.
Rank is that count: 0 for a scalar, 1 for a vector, 2 for a matrix, 3 or more for a full tensor. The same rules of addition and multiplication apply at every level.
Tensors are not a new kind of object. They are the general case that includes everything else.
The aurora is sunlight’s most beautiful reminder that Earth has a magnetic shield.
When charged particles from the Sun reach Earth, our magnetic field guides many of them toward the polar regions. There, collisions with oxygen and nitrogen excite the atoms, which release light as they return to lower energy states.
Oxygen produces the familiar green glow at 557.7 nm, while red auroras often come from oxygen at 630.0 nm.
What looks like light dancing across the sky is actually atomic energy being released, one photon at a time.
A cube has 8 vertices, 12 edges, and 6 faces.
8 − 12 + 6 = 2.
A tetrahedron has 4 vertices, 6 edges, and 4 faces.
4 − 6 + 4 = 2.
Euler discovered that every convex polyhedron obeys the same relationship:
V − E + F = 2.
The shape can change completely, but this number remains unchanged.