Use ChatGPT to learn math at your own peril.
I recently started a self study of linear and abstract algebra. I tried to use chatgpt in order to look up definitions. Look what I got1 :
Prompt 1: Define linear operator:
Definition of a Linear Operator
A linear operator is a mapping T between two vector spaces that preserves vector addition and scalar multiplication.
Formal Definition
Let V and W be vector spaces over a field {F}. A function
T: V -> W is called a linear operator if it satisfies the following properties for all u, v in V and all scalars c in {F}:
- Additivity: T(u + v) = T(u) + T(v).
- Homogeneity (Scalar Multiplication): T(cu) = cT(u).
Special Case: Endomorphism
If V = W, meaning the operator maps a vector space to itself, T is called a linear transformation or an endomorphism.
Matrix Representation
In the context of finite-dimensional vector spaces, a linear operator can be represented by a matrix, and applying the operator corresponds to multiplying the matrix by a vector.
Prompt 2: Describe the relationship between a linear transformation and a linear operator.
Relationship Between Linear Transformation and Linear Operator
A linear transformation is a function between two vector spaces that preserves vector addition and scalar multiplication.
T: V -> W
Linear Operator
A linear operator is a linear transformation that maps a vector space to itself:
T: V -> V
Key Relationship
- Every linear operator is a linear transformation.
- Not every linear transformation is a linear operator because a transformation may map between different vector spaces.
Conclusion
A linear operator is a special case of a linear transformation where the domain and codomain are the same vector space.
So is a linear operator a special case of a linear transformation or vice versa? Chatgpt gave me contradicting answers. Poking around other sources has led me to believe that the two terms are used interchangeably.
Here's wolframlpha's definition of a linear transformation. For the time being I'll be using that website as the resource for math related definitions. I noticed that they cite the contributor. Thanks Todd Rowland
- I had to fiddle with the latex response to make this somewhat html friendly, chatgpt gave me quality latex output that led me to have false confidence in the validity of that output [↩]