Basis Vectors
- a set 𝐵 of vectors in a vector space 𝑉 is called a basis if it is a linearly independent spanning set, i.e.:
- every element of 𝐵 is linearly independent
- every element of 𝑉 is a FINITE linear combination of the elements of 𝐵 (i.e. 𝐵 spans 𝑉)
- vector space is to the basis vector as function space is to the basis function
- is a (0,1)-tensor (except for the dual basis covectors)
Basis Vectors - Types
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Basis Vectors - Linear Transformation Intuition
Understanding how a linear transformation 𝑇 transforms standard basis vectors 𝑖 and 𝑗
For example, given the following linear transformation 𝑇:
and also the following basis vectors 𝑖 and 𝑗 (in this example 𝑖 and 𝑗 are the standard basis vectors):
Taking the transformation of 𝑇 on 𝑖 and 𝑗:
This is what it looks like graphically
Understanding how a linear transformation 𝑇 transforms an arbitrary vector 𝑣, where 𝑣 is expressed by basis vectors 𝑖 and 𝑗
For example, given the following linear transformation 𝑇, basis vectors 𝑖 and 𝑗, and vector 𝑣:
Vector 𝑣 can be decomposed into its constituent basis vectors:
Taking the transformation of 𝑇 on 𝑣 is the same as taking the transformations of the basis vectors 𝑖 and 𝑗 individually:
This is what it looks like geometrically
