Questions and Answers
What is the definition of onto?
What is the definition of one to one?
What does Theorem 2 state about a linear transformation T?
T maps Rn to Rm onto if and only if the columns of A span Rm.
What does Theorem 1 state about a linear transformation T?
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Study Notes
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.
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Description
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.