Determine if the columns of the matrix A span R2.
A =

3	1	0	

12	−4	−1
The columns span R2.
The columns do not span R2.

To determine if the columns of the matrix \( A \) span \( \mathbb{R}^2 \), we need to check if the matrix has two linearly independent columns. If the columns are linearly independent, they will span \( \mathbb{R}^2 \). The matrix \( A \) is given as:

\[
A = \begin{pmatrix}
3 & 1 & 0 \\
12 & -4 & -1
\end{pmatrix}
\]

### Step 1: Look at the number of columns
The matrix has 3 columns, but we only need two columns to check if they span \( \mathbb{R}^2 \). We can analyze the first two columns, because if two columns are linearly independent, then they will form a basis for \( \mathbb{R}^2 \).

So, we extract the first two columns:

\[
B = \begin{pmatrix}
3 & 1 \\
12 & -4
\end{pmatrix}
\]

### Step 2: Compute the determinant
For two vectors (or columns) to be linearly independent, the determinant of the matrix formed by these vectors should be non-zero. We will compute the determinant of \( B \):

\[
\text{det}(B) = (3)(-4) - (1)(12) = -12 - 12 = -24
\]

Since the determinant is non-zero (\(-24\)), the first two columns are linearly independent, and they span \( \mathbb{R}^2 \).

### Conclusion:
The columns of matrix \( A \) **span \( \mathbb{R}^2 \)**.

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Would you like more details, or have any questions?

Here are 5 related questions you could explore:

1. What does it mean for a set of vectors to span a vector space?
2. How do you determine if a set of vectors is linearly independent?
3. What is the geometric interpretation of two vectors spanning \( \mathbb{R}^2 \)?
4. How can the rank of a matrix help determine the dimension of the span?
5. What happens if the determinant of a 2x2 matrix is zero?

**Tip:** When checking if columns of a matrix span a space, always consider the matrix's rank—it's a key indicator of independence and dimensionality.

Determine if the columns of the matrix A span R2.
A =
3	1	0
12	−4	−1
The columns span R2.
The columns do not span R2.

Learn how to determine if the columns of matrix A span R2 by checking the linear independence of the first two columns. This problem uses the determinant of a 2x2 matrix and the concept of linear independence in vector spaces. Suitable for undergraduate students studying linear algebra.

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