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.