Basis vector

As explained in Geometry for Programmers

A basis vector is a vector in a basis whose scaled combinations represent vectors in the associated vector space.

A basis vector is one of the reference vectors used to build other vectors through addition and scalar multiplication. In two dimensions, i = (1, 0) and j = (0, 1) form the standard orthonormal basis, so a = (0.5, 1.5) can be written as 0.5i + 1.5j. The coefficients are the vector's coordinates in that basis. section 2.1.2

Keep these practical distinctions in view:

• Tuple notation and a weighted sum of basis vectors describe the same vector in the standard basis: (2, 3, 4) = 2i + 3j + 4k. section 9.1
• The standard basis is orthonormal because its vectors are perpendicular and have length 1; other bases may be nonorthogonal or non-normalized. section 2.1.2
• Changing the basis changes the reference directions and the coefficients used to represent a vector, while keeping the vector operations available. section 2.1.2

Representing a 3D vector in SymPy

SymPy models vectors as weighted sums of basis vectors in a coordinate system. After creating N with CoordSys3D('N'), write the vector (0, -1, 2) as 2*N.k - N.j. This representation also supports operations such as dot products. section 9.2.2

Figure 2.6 A point vector a written in orthonormal basis as 0.5i + 1.5j

Visualizes a point vector as 0.5i + 1.5j in an orthonormal basis, connecting coefficients with geometric direction and length. section 2.1.2

How Geometry for Programmers covers it

How deep it goes

Defines basis vectors through 2D and 3D decompositions, explains orthonormality, and stops short of formal basis theory.

Examples

Uses i and j in 2D, i, j, and k in 3D, geometric figures, and SymPy coordinate-system listings.

What it assumes

Assumes basic vectors, addition, scalar multiplication, and coordinate tuples while introducing basis notation from those operations.

What you can do after

Apply it via tuple notation, weighted basis-vector expressions, and SymPy coordinate systems.

How it connects

Connects basis vectors with vectors, scalar multiplication, coordinate systems, orthonormal bases, and dot products.

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Related concepts

• vectorkind of · A vector is an ordered collection of components representing an element of a vector space.
• coordinate systempart of · A coordinate system is a framework for specifying the positions of points or other geometric entities using numerical coordinates relative to defined references, such as an origin and axes.
• basispart of · A basis is a linearly independent set of vectors whose linear combinations represent every vector in a vector space.

This explainer was generated by Manning's knowledge graph from the text of Geometry for Programmers. The section links point to the parts of the book it draws on.