Podcast
Questions and Answers
What does it mean for the function Ax=b to be onto?
What does it mean for the function Ax=b to be onto?
The function Ax=b is onto Rm if A has a pivot position in every row.
The function Ax=b is onto Rm if A has a pivot position in every row.
True
What is a condition for a function Ax=b to be one-to-one?
What is a condition for a function Ax=b to be one-to-one?
A has a pivot position in every column.
Which of the following statements are true for the function Ax=b being one-to-one? (Select all that apply)
Which of the following statements are true for the function Ax=b being one-to-one? (Select all that apply)
Signup and view all the answers
If the columns of A are linearly dependent, then the function Ax=b is one-to-one.
If the columns of A are linearly dependent, then the function Ax=b is one-to-one.
Signup and view all the answers
Study Notes
Onto Functions
- The function Ax = b is considered onto when for every vector b in R^m, there exists at least one solution.
- A crucial characteristic of onto functions is that each b can be expressed as a linear combination of the columns of matrix A.
- To determine if A is onto, verify that its columns span the entire space R^m.
- A must have a pivot position in every row for the function to be classified as onto.
One-to-One Functions
- The function Ax = b is classified as one-to-one when there is at most one solution for each vector b.
- This implies that the equation Ax = 0 will only have the trivial solution, meaning the only solution is the zero vector.
- One-to-one functions are characterized by the linear independence of the columns of A.
- A must have a pivot position in every column, ensuring no redundancy among the columns, for the function to be one-to-one.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
This quiz focuses on understanding onto functions in linear algebra, specifically through the equation Ax=b. Each question examines key concepts such as spanning, pivot positions, and the properties of matrices related to onto mappings. Test your knowledge on these essential topics!
