Linear Algebra: Onto and One-to-One

Questions and Answers

What is the definition of onto?

A mapping from Rn to Rm that does not cover the entire co-domain.

A mapping where some vectors in the co-domain do not have pre-images in the domain.

A mapping where every vector in the co-domain is the image of at least one vector in the domain. (correct)

A mapping where each vector in the domain is the image of only one vector in the co-domain.

What is the definition of one to one?

A mapping where every vector in the domain is the image of at least one vector in the co-domain.

A mapping where no vectors in the co-domain are left without pre-images.

A mapping where each vector in the co-domain is the image of at most one vector in the domain. (correct)

A mapping that can have multiple pre-images for a specific vector in the co-domain.

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?

<p>T is one to one if and only if T(x) = 0 has only the trivial solution.</p> Signup and view all the answers

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.

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.