Linear Algebra: Onto Functions

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Questions and Answers

What does it mean for the function Ax=b to be onto?

The equation Ax=0 has only the trivial solution.
Each b in Rm is a linear combination of the columns of A. (correct)
A has a pivot position in every row. (correct)
The columns of A span Rm. (correct)

The function Ax=b is onto Rm if A has a pivot position in every row.

True (A)

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)

<p>The columns of A are linearly independent. (A), The equation Ax=b has at most one solution for every b. (C), The equation Ax=0 has only the trivial solution. (D)</p> Signup and view all the answers

If the columns of A are linearly dependent, then the function Ax=b is one-to-one.

<p>False (B)</p> Signup and view all the answers

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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.