Lecture 7 defined the coordinate vector of a vector with respect to a basis, and established that it is well-defined (unique representation proposition). This write-up asks the next natural question: is the assignment a linear transformation? The answer is yes — and more is true. The map is also a bijection, making it an isomorphism of vector spaces. The consequence is that every finite-dimensional vector space is, from the perspective of linear algebra, the same as .
1. The coordinate map
Fix a vector space of dimension and a basis of .
Definition — Coordinate map
The coordinate map associated to is the function
Explicitly: if , then .
This is well-defined: the unique representation proposition (link) guarantees that the scalars exist and are uniquely determined by , so the map has no ambiguity.
2. The coordinate map is linear
Theorem
For any basis of a vector space , the coordinate map is a linear transformation.
Proof
We verify the two linearity conditions.
Setup. Let and write their unique basis expansions:
By definition, and .
Condition 1: .
Add the two expansions:
This is already the basis expansion of — and by uniqueness of representation, it is the only one. Therefore:
Condition 2: for all .
Scale the expansion of :
Again this is the unique basis expansion of , so:
What the proof actually says. Adding two vectors in corresponds to adding their coefficient lists in ; scaling a vector in corresponds to scaling its coefficient list. The coordinate map translates the abstract operations of into the concrete componentwise operations of . This is exactly what linearity means.
3. Worked example
Let , the space of polynomials of degree at most . We work with two different bases and compare.
Standard monomial basis
Take . Every polynomial has coordinate vector
Let and . Then:
Checking linearity., so . And indeed .
For scalar multiplication: , so .
A non-standard basis
Take . To find for , solve:
Expanding the left side: . Matching coefficients:
So . The same polynomial , a different coordinate vector — reflecting a different choice of “grid” on .