Linear Combinations

Linear Algebra for Machine Learning · Axiom Academy

Expressing vectors as weighted sums A linear combination of vectors v₁, v₂, ..., vₙ is any vector of the form: where c₁, c₂, ..., cₙ are scalars (coefficients). Express v = [5, 4] as a linear combination Given: v₁ = [1, 2] and v₂ = [2, 1] Find: c₁ and c₂ such that v = c₁v₁ + c₂v₂ We've expressed [5, 4] as 1·[1, 2] + 2·[2, 1]. This is a linear combination! Question 1: Linear Combinations Is the zero vector a linear combination of any non-zero vector v? The span of a set of vectors is the set of all possible linear combinations. What is the span of [1, 0] and [0, 1]?

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