True or false: If matrix A is n×n and the columns of matrix A are linearly dependent, then the columns cannot span ℝn. 

**True.** If the columns of an \( n \times n \) matrix \( A \) are linearly dependent, then they do not span \(\mathbb{R}^n\).

Here's an explanation:

1. **Linear Dependence and Span:** For a set of vectors to span \(\mathbb{R}^n\), they must be linearly independent. Linear independence means that no vector in the set can be written as a linear combination of the others. 

2. **Matrix Columns and Rank:** The rank of a matrix is the maximum number of linearly independent columns (or rows) it has. For an \( n \times n \) matrix \( A \) to have columns that span \(\mathbb{R}^n\), the rank of \( A \) must be \( n \). 

3. **Linear Dependence Implications:** If the columns of \( A \) are linearly dependent, then the rank of \( A \) is less than \( n \). Consequently, there are not enough linearly independent vectors to span an \( n \)-dimensional space.

Therefore, if the columns of matrix \( A \) are linearly dependent, they cannot span \(\mathbb{R}^n\).

Would you like more details or have any questions?

Here are some related questions for further understanding:

1. How do you determine if a set of vectors is linearly independent?
2. What is the rank of a matrix, and how is it calculated?
3. Can a set of vectors span a space if they are not linearly independent? Why or why not?
4. What is the relationship between the determinant of a matrix and the linear independence of its columns?
5. How do the concepts of eigenvalues and eigenvectors relate to the span and linear dependence of a matrix?

**Tip:** Remember that the rank of a matrix provides crucial information about the linear independence of its columns and the dimensions of the space they span.

True or false: If matrix A is n×n and the columns of matrix A are linearly dependent, then the columns cannot span ℝn.

Explore the relationship between linear dependence and the span of ℝn in an n×n matrix. This detailed explanation covers the concepts of rank, linear independence, and their role in determining the span of a vector space. Suitable for undergraduate students studying linear algebra.

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