How deeply does Math and Architectures of Deep Learning cover basis?

Defines spanning, dimensionality, nonuniqueness, orthogonality, and linear independence, while stopping short of computational implementation.

Basis

As explained in Math and Architectures of Deep Learning

A vector in a spanning set that provides a coordinate direction for representing every vector in the space.

A basis is a set of vectors that spans a vector space, allowing every vector in that space to be expressed as a linear combination of the basis vectors. Geometrically, basis vectors provide coordinate directions for describing points in high-dimensional spaces. section 2.9 section 2.9.3

A basis has several defining properties:

Exactly n basis vectors are needed to span an n-dimensional space such as ℝⁿ. section 2.9.3
Basis vectors need not be orthogonal; any linearly independent pair forms a basis for ℝ². section 2.9.3
The familiar Cartesian coordinate axes are one basis, but bases are not unique. section 2.9.3

Coordinate directions in ℝ²

The two standard coordinate-direction vectors form a basis for ℝ²: every two-dimensional vector can be represented as a linear combination of them. Replacing them with any two linearly independent vectors gives another basis, even when the directions are not perpendicular. section 2.9.3

How Math and Architectures of Deep Learning covers it

Cover of Math and Architectures of Deep Learning

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Related concepts

order of continuityconfigures · The order of continuity is the highest derivative order through which a function and its derivatives remain continuous.
orthogonalitycontrasts with · Orthogonality is the relationship between vectors, directions, or states whose inner product is zero, corresponding to perpendicularity in Euclidean space.
computational basisincludes · The computational basis is the standard basis of quantum states labeled by definite binary-string measurement outcomes.

This explainer was generated by Manning's knowledge graph from the text of Math and Architectures of Deep Learning. The section links point to the parts of the book it draws on.