Linear Algebra

Everything about AM 21b 2017

Everything about AM 21b 2018

 I’ll post sections of notes on linear algebra when they are ready.

• Linear equation
• Scalar
• Vector
• Linear map
• Linear form
• Bilinear form
• Operator
• Tensor
• Affine space
• Euclidean space
• Minkowski space
• Hilbert spacce

Applications of linear algibra

• Sankey diagram
• Input-output economics

Background from basic mathematics

• Set
• Cartesian product
• Relation
• Map
• Group
• Number

Being linear within a set defines a vector space.  Being linear between sets defines a linear map.  A vector itself is a linear map. A linear map is an element of a new vector space. We map the new vector space to some other vector spaces—that is, we map the maps. So goes the never-ending story of linear algebra.

We call a one-dimensional vector space a scalar set.  We specify the linear algebra of a world by a number field, a vector space, and a collection of scalar sets.  We form linear maps between the vector space and scalar sets, as well as linear maps between the linear maps.  The scalars, vectors, and linear maps are collectively called tensors.

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