The Redundant Vectors Get Left Behind. That's Fine | Linear Algebra | Vector Spaces | Dogmathic

Span and Basis of a Subset of R⁴ | Linear Algebra #Dogmathic #LinearAlgebra #BasisAndSpan #GATEMaths The span and basis of a set of vectors are two things people confuse until the day they stop confusing them, which is usually right around the time row reduction finally clicks. This video works through a concrete example: five vectors, each with four components, living in R⁴. We find both. The approach is hands-on. No shortcuts. We set up a matrix whose columns are the five vectors, then run reduced row echelon form on it from start to finish, fractions included, unfortunately. Along the way, we talk about what pivots are telling you, why you'd want ones in some places and zeros in others, and how adding rows versus scaling rows each accomplish a different kind of cleanup. The span turns out to be generated by just three of the five vectors, V1, V2, and V4. The other two, V3 and V5, are linear combinations of those three. They're not useless, they're just redundant. There's a difference. Once the span is settled, we run RREF again on the reduced set to confirm the basis, and the identity block that falls out makes the linear independence obvious. It's a good problem for anyone who has done row reduction mechanically but never quite connected it to what span and basis actually mean geometrically and algebraically. The fractions are unpleasant. The conclusion is clean. Topics covered: span of a set, basis of a vector space, reduced row echelon form, RREF, Gaussian-Jordan elimination, linear independence, pivot columns, linear combination, row operations, R⁴, matrix column space, linearly dependent vectors, vector space, linear algebra https://dogmathic.com/ matherssen(at)gmail.com Properties and Concepts Used: Reduced Row Echelon Form (RREF) Gaussian-Jordan Elimination Pivot columns and pivot positions Linear independence and dependence Span of a set of vectors Basis of a vector space Row operations: scalar multiplication and row addition Linear combination of vectors Column space of a matrix Identifying redundant vectors via RREF R⁴ as a vector space Matrix representation of a set of vectors Reciprocal multiplication to clear fractions in rows Reading basis vectors from original (pre-reduction) matrix    • Two Vectors Walk Into a Span. One Doesn't ...      • The Redundant Vectors Get Left Behind. Tha...      • Why Do Zeros Make Your Life So Much Easier...      • This Matrix Destroys Every Vector Except T...      • Linear Algebra      • Topology (Point-Set)      • Real Analysis      • Abstract Algebra   Chapters: 0:00 Introduction and problem setup 1:08 Setting up the matrix A 1:43 First row operations — getting the leading 1 5:03 Clearing fractions, scaling rows 9:29 Closing in on RREF — last pivot 13:02 Reading the span from pivot columns 15:05 Finding the basis (RREF round two) 18:11 Wrapping up https://dogmathic.com/ matherssen(at)gmail.com #Dogmathic #LinearAlgebra #BasisAndSpan #GATEMaths