Can an m×n matrix A have a nullity of n where n>m ?

No, an \( m \times n \) matrix \( A \) cannot have a nullity of \( n \) where \( n > m \).

### Here's why:
- **Nullity** is defined as the dimension of the null space of the matrix \( A \). The null space consists of all vectors \( \mathbf{x} \) such that \( A\mathbf{x} = \mathbf{0} \).
- For an \( m \times n \) matrix \( A \), the rank-nullity theorem states:
  \[
  \text{Rank}(A) + \text{Nullity}(A) = n
  \]
  where:
  - \( \text{Rank}(A) \) is the rank of the matrix, or the dimension of the column space.
  - \( \text{Nullity}(A) \) is the dimension of the null space.

- Since \( n > m \), the maximum possible rank of \( A \) is \( m \) (because there are only \( m \) rows). Therefore, the rank of \( A \) can be at most \( m \), and the nullity can be at least \( n - m \).

- However, the nullity \( \text{Nullity}(A) \) cannot be equal to \( n \), because this would imply that the rank \( \text{Rank}(A) \) is \( 0 \). This would mean \( A \) has no linearly independent columns, and hence \( A \mathbf{x} = \mathbf{0} \) for every vector \( \mathbf{x} \), which is not possible unless \( A \) is the zero matrix (in which case the nullity would be \( n \) only if \( m = 0 \), which is not the case here since \( m > 0 \)).

Thus, it's impossible for an \( m \times n \) matrix to have a nullity of \( n \) when \( n > m \).

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. What is the significance of the rank-nullity theorem in linear algebra?
2. How does the concept of rank relate to the linear independence of the columns of a matrix?
3. Can an \( m \times n \) matrix have a null space with dimension \( m \) if \( m \neq n \)?
4. How does the rank of a matrix affect the solution to a system of linear equations \( A\mathbf{x} = \mathbf{b} \)?
5. What is the geometric interpretation of the null space and column space of a matrix?

**Tip:** Understanding the rank-nullity theorem is crucial for grasping the relationship between the solutions to a linear system and the structure of a matrix.

Explore why an m×n matrix A cannot have a nullity of n where n > m through detailed explanations of the Rank-Nullity Theorem and matrix properties. Gain insights into linear algebra concepts crucial for understanding matrix dimensions and solutions.

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