Reverse Engineer the Linear Combination | #mathmachinelearning | #linearalgebra | #dogmathic

#LinearAlgebra #MathForML #MachineLearning #VectorSpaces #MLMath Seven Steps of Row Reduction to Find Three Numbers. Math Is Humbling. Linear combinations and Gauss-Jordan elimination show up together here because they always show up together. This is Exercise 2.11 from Mathematics for Machine Learning, and the job is straightforward: given a target vector y and three other vectors, find the scalars λ₁, λ₂, λ₃ that make the combination work. The definition of a linear combination fits in one line. Definition 2.11, page 40: scale each vector, add them up, get something new. That is the whole thing. The exercise runs it backwards. We have the result (y), we have the vectors, and we need to find the scalars. Which means setting up a system of three equations in three unknowns. Which means an augmented matrix. Which means Gauss-Jordan, whether you wanted it or not. Seven elimination steps: zero out the first column, zero out the second, scale the third pivot, then back-substitute up through row one. The matrix grinds. The answer reads off cleanly at the end: λ₁ = −6, λ₂ = 3, λ₃ = 2. So y = −6x₁ + 3x₂ + 2x₃. Short video. Boring in the best way. The kind of problem that makes linear algebra feel like plumbing, which is not entirely wrong. Linear combinations are the foundation of everything that comes later in this subject. Topics covered: linear combination, Gauss-Jordan elimination, augmented matrix, row reduction, pivot, back substitution, systems of linear equations, vector spaces, scalars, Mathematics for Machine Learning, linear algebra, MML exercise, matrix row operations, RREF, discrete math Support Dogmathic https://ko-fi.com/dogmathic https://dogmathic.com/ matherssen(at)gmail.com https://youtu.be/sUrdV31_084 https://youtu.be/qXZ4sXgJFGk https://youtu.be/rCqL-ZhAK5g https://youtu.be/0msRpIW2kAw https://youtu.be/_X9D7YS9oEQ https://www.youtube.com/playlist?list=PLm90IN9RVLf-hf1BPIxN6lW2oqfP8a4Mq https://www.youtube.com/playlist?list=PLm90IN9RVLf-8Ht5hWSodKFwOOG-FWxBx https://www.youtube.com/playlist?list=PLm90IN9RVLf-W0SGnXjWpP3r8wm6Vq1mn https://www.youtube.com/playlist?list=PLm90IN9RVLf9hn9po3pPHzK540MCY6XMY https://www.youtube.com/playlist?list=PLm90IN9RVLf-mwflhqqrGCQHUWDWgomcB Properties and Concepts Used: Linear combination (Definition 2.11, MML p. 40) Augmented matrix construction (stacking column vectors) Gauss-Jordan elimination Forward elimination (zeroing entries below pivots) Back substitution (zeroing entries above pivots) Row scaling (multiplying a row by a scalar) Pivot identification Reduced row echelon form (RREF) Systems of linear equations (3×3) Vector representation in Rⁿ Scalar multiplication on vectors Vector addition Unique solution existence (full-rank system) Chapters: 0:00 Introduction and book context 0:22 Definition 2.11: linear combinations 0:54 Building the augmented matrix 1:18 Forward elimination (steps 1-4) 2:13 Back substitution (steps 5-7) 2:59 Reading off λ₁, λ₂, λ₃ 3:13 Final answer: y = −6x₁ + 3x₂ + 2x₃ #LinearAlgebra #MathForML #MachineLearning #VectorSpaces #MLMath