Linear Algebra: Onto and One-to-One

Definitions of Onto and One to One

• A mapping from Rⁿ to Rᵐ is onto when every vector in the co-domain Rᵐ is represented as the image of at least one vector from the domain.
• This implies that the image of the transformation encompasses the entire co-domain.
• A mapping from Rⁿ to Rᵐ is one to one when each vector in the co-domain Rᵐ corresponds to at most one vector from the domain Rⁿ.
• This means no two distinct vectors in the domain map to the same vector in the co-domain.

Theorem 2 - Conditions for Onto

• A linear transformation T: Rⁿ → Rᵐ is onto if the columns of the standard matrix A for T span the entire co-domain Rᵐ.
• The spanning of Rᵐ by the columns of A occurs if there is a pivot in every row of the matrix A.

Theorem 1 - Conditions for One to One

• A linear transformation T: Rⁿ → Rⁿ is one to one if the equation T(x) = 0 has only the trivial solution, meaning the only solution is the zero vector.
• This property is equivalent to the columns of matrix A being linearly independent.
• Linear independence of the columns is confirmed by having a pivot in every row of matrix A.

This quiz focuses on the key concepts of onto and one-to-one mappings in linear algebra. Explore the definitions and theorems that describe these fundamental properties of functions. Perfect for reinforcing your understanding of mapping functions in vector spaces.