Two Vectors Walk Into a Span. One Doesn't Make It Out | Linear Algebra | Vector Spaces | Dogmathic
Are These Vectors in the Span? Let's Find Out #Dogmathic #LinearAlgebra #MathProof #VectorSpaces #GATEMaths
Span and linear independence are the beating heart of linear algebra. A span is a set, in this case, a plane through the origin in R3, and the question is simple: does a given vector live in it, or not. Most students stare at this and freeze. The answer, it turns out, is in the augmented matrix the whole time.
We take X = span{[1,0,2], [0,1,−1]} and test two vectors: v = [2,3,1] and u = [1,1,0]. The move is to reframe membership as a system of equations, Ax = b, where A holds the spanning vectors as columns and b is the candidate. If the system is consistent, the vector is in the span. If it isn't, the vector has nowhere to live on that plane.
For v, the system works out. s = 2, t = 3, and the third row checks out cleanly: 4 − 3 = 1. v is in X. For u, s = 1, t = 1, and the third row asks whether 2(1) − 1 = 0. It doesn't. One is not zero. u is not in X, and that's the end of it.
Linear algebra is full of moments where the formalism feels heavy and the answer turns out to be a single row of arithmetic. This is one of those moments.
If you've been testing vectors by hand, try this: what's the smallest change you could make to u to put it in X?
Topics covered: span, linear span, vector space membership, linear combination, systems of linear equations, matrix-vector multiplication, Ax=b, consistency of linear systems, R3, plane through origin, basis vectors, linear algebra, row reduction, spanning set, vector space, linear independence
https://dogmathic.com/
matherssen(at)gmail.com
• 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
Properties and Concepts Used:
Span of a set of vectors
Linear combination
Vector membership in a span
Matrix-vector multiplication (Ax = b formulation)
Systems of linear equations (consistency)
Row-by-column multiplication
R3 as three-dimensional Euclidean space
Plane through the origin in R3
Augmented matrix (mentioned, not used)
Reduced row echelon form (mentioned, not used)
Parametric solution variables (s and t)
Back-substitution / direct substitution to verify consistency
Chapters:
0:00 Introduction & Setup
0:36 What Is the Span? (Plane in R3)
1:51 The Strategy: Reframe as Ax = b
4:44 Testing v = [2,3,1]
6:41 v Is in X — Here's the Proof
7:21 Testing u = [1,1,0]
8:26 u Is Not in X — The Third Row Decides
9:01 Conclusion & Summary
#Dogmathic #LinearAlgebra #MathProof #VectorSpaces #GATEMaths