https://file.aiquickdraw.com/mathbot/file/1725834505245ajfcv6m7.jpg

The question in the image asks:

"If matrix \( A \) is \( n \times n \) and the columns of matrix \( A \) are linearly dependent, then the columns cannot span \( \mathbb{R}^n \)."

The answer marked is "True."

### Explanation:

- **Linear Dependence**: The columns of a matrix are linearly dependent if at least one of the columns can be written as a linear combination of the others.

- **Spanning \( \mathbb{R}^n \)**: The columns of an \( n \times n \) matrix span \( \mathbb{R}^n \) if they form a basis for \( \mathbb{R}^n \), meaning the matrix must have full rank (rank \( n \)).

If the columns are linearly dependent, the matrix does not have full rank, meaning it cannot span \( \mathbb{R}^n \). Therefore, the statement is **True**.

Would you like further details on this topic or have any questions?

Here are five related questions:

1. What is the rank of a matrix and how is it determined?
2. How can you check if a set of vectors spans a vector space?
3. What is the relationship between the determinant of a matrix and its linear independence?
4. How do you find the basis of the column space of a matrix?
5. What is the significance of the null space in understanding the linear dependence of a matrix?

**Tip**: Remember, if the determinant of a square matrix is zero, the matrix is singular, meaning its columns are linearly dependent, and it does not span the entire space.

If matrix A is n x n and the columns of matrix A are linearly dependent, then the columns cannot span R^n.

This problem explores the linear dependence of columns in an n x n matrix and their ability to span R^n. It covers key linear algebra concepts, such as matrix rank, the span of vectors, and the implications of linear dependence. Suitable for university-level students.

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