Questions and Answers
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.
True
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)
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If the columns of A are linearly dependent, then the function Ax=b is one-to-one.
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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.
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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!
