Ep 14: When 'Vector' Stops Meaning 'Arrow' | Vector Spaces and Subspaces

For thirteen episodes a vector has been an arrow, or a list of numbers. Now we make the leap that gives linear algebra its astonishing reach. We forget what vectors are made of and remember only what we can do to them: add two of them, and scale one by a number. If those two operations obey eight simple axioms — closure, commutativity, an additive identity and inverses, and the distributive and compatibility laws — then we call the whole collection a vector space, and everything we proved for arrows in R-n comes along for free. The surprise is who joins the club. Polynomials are vectors: you can add them and scale them. Matrices of a fixed size are vectors. Functions are vectors. Infinite sequences are vectors. Each is a perfectly good vector space, even though none of them looks like an arrow. Then we zoom in: a subspace is a vector space living inside a larger one, and a single three-part test decides whether a subset qualifies. We will see that the span of any set of vectors is automatically a subspace, that every space contains a smallest subspace — just the zero vector — and a largest one, the whole space itself. 📚 Continues in: → 'Linear Independence, Basis, and Dimension' — once we know what a vector space is, we ask how few vectors are needed to build all of it, and how many are too many. That count is the dimension. 🎓 Best for: A first or second college linear algebra course, students in engineering / CS / data science / physics, anyone heading into machine learning, and self-learners who want one coherent course instead of scattered clips. — QED Realized — Clean explanations of the ideas that take a real lecture to land. 🔔 Subscribe so you don't miss the next one:    / @qedrealized   #LinearAlgebra #Math #Mathematics #VectorSpaces #Subspaces #AbstractAlgebra — Tags (for search) — vector space, vector space axioms, eight axioms of a vector space, abstract vector space definition, vector space examples, polynomials as a vector space, space of matrices, function space, sequence space, subspace definition, subspace test, three part subspace test, closure under addition, closure under scalar multiplication, contains the zero vector, span is a subspace, zero subspace, trivial subspace, whole space subspace, examples of subspaces, lines and planes through the origin, grassmann peano vector space history, linear algebra abstract structures tutorial ⏱ Chapters 0:00 Forget the arrow, keep the operations 0:45 The complete definition of a vector space 1:36 P_n, polynomials of degree at most n 2:24 Three more spaces with no arrows 3:14 A vector space living inside another 4:02 Three checks, and you are done 4:54 The most reliable subspace factory 5:42 \{\mathbf 0\} and V itself 6:31 Why it is a subspace, and a parallel plane is not 7:21 From the origin up to all of space 8:08 Lines through the origin versus shifted lines 9:00 Which sets of functions are subspaces? 9:59 A subspace inside the space of matrices 10:52 Grassmann, Peano, and the rise of abstraction 11:38 Recap 12:36 Thanks for watching