Basis and Dimension

Linear Algebra for Machine Learning · Axiom Academy

Minimal spanning sets and space structure A basis is a minimal set of independent vectors that spans the entire space. A set of vectors v₁, v₂, ..., vₖ is a basis if: • They are linearly independent • Their span is the entire space Every vector in the space can be uniquely written as a combination of basis vectors. The simplest basis vectors are standard basis vectors (unit vectors): In ℝ²: e₁ = [1, 0] and e₂ = [0, 1] In ℝ³: e₁ = [1, 0, 0], e₂ = [0, 1, 0], e₃ = [0, 0, 1] In ℝⁿ: n vectors with a 1 in position i and 0s elsewhere Every vector as a combination: [3, 5] = 3[1, 0] + 5[0, 1] The coefficients are just the coordinates! The dimension of a space is the number of vectors in any basis. Key fact: All bases of the same space have the same number of vectors. Dimension tells us how many independent features are needed to describe vectors in that space. It's a fundamental property!

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