https://file.aiquickdraw.com/mathbot/file/17293487478039wm80i2j.jpg

Let's analyze the two questions presented in the image.

### **Question 19: Linear Independence of Vectors using Determinant**
You are asked to determine which set of vectors is **linearly independent**. Two vectors \(\vec{v}_1\) and \(\vec{v}_2\) in \(\mathbb{R}^2\) are linearly independent if the determinant of the matrix formed by these vectors is non-zero. The matrix can be written as:

\[
A = \begin{bmatrix}
\vec{v}_1 & \vec{v}_2
\end{bmatrix} = 
\begin{bmatrix}
v_{11} & v_{12} \\
v_{21} & v_{22}
\end{bmatrix}
\]

The determinant of this 2x2 matrix is calculated as:

\[
\text{det}(A) = v_{11} \cdot v_{22} - v_{12} \cdot v_{21}
\]

Let's calculate the determinant for each option:

- **Option 1: \(\vec{v}_1 = (2, 1)\), \(\vec{v}_2 = (5, 4)\)**

\[
\text{det} = (2)(4) - (1)(5) = 8 - 5 = 3 \quad (\text{non-zero, linearly independent})
\]

- **Option 2: \(\vec{v}_1 = (3, 2)\), \(\vec{v}_2 = (6, 4)\)**

\[
\text{det} = (3)(4) - (2)(6) = 12 - 12 = 0 \quad (\text{zero, linearly dependent})
\]

- **Option 3: \(\vec{v}_1 = (1, -1)\), \(\vec{v}_2 = (4, -4)\)**

\[
\text{det} = (1)(-4) - (-1)(4) = -4 + 4 = 0 \quad (\text{zero, linearly dependent})
\]

**Answer for Question 19: Option 1** is correct because it has a non-zero determinant, meaning the vectors are linearly independent.

---

### **Question 20: Basis of a Vector Space using Determinant**
For vectors to form a basis for \(\mathbb{R}^2\), they must be linearly independent. Therefore, the determinant of the matrix formed by the two vectors must be non-zero.

Let's calculate the determinant for each set:

- **Option 1: \(\vec{v}_1 = (1, 5)\), \(\vec{v}_2 = (2, 10)\)**

\[
\text{det} = (1)(10) - (5)(2) = 10 - 10 = 0 \quad (\text{zero, not a basis})
\]

- **Option 2: \(\vec{v}_1 = (1, 2)\), \(\vec{v}_2 = (3, 6)\)**

\[
\text{det} = (1)(6) - (2)(3) = 6 - 6 = 0 \quad (\text{zero, not a basis})
\]

- **Option 3: \(\vec{v}_1 = (2, 1)\), \(\vec{v}_2 = (1, 2)\)**

\[
\text{det} = (2)(2) - (1)(1) = 4 - 1 = 3 \quad (\text{non-zero, forms a basis for \(\mathbb{R}^2\)})
\]

**Answer for Question 20: Option 3** is correct because it has a non-zero determinant, meaning these vectors form a basis for \(\mathbb{R}^2\).

---

Would you like more details on any of these questions?

### Relative Questions:
1. How do you find the determinant of larger matrices, such as 3x3?
2. What does it mean geometrically for vectors to be linearly dependent?
3. Can vectors in \(\mathbb{R}^3\) be linearly independent if there are more than 3 vectors?
4. How does linear independence relate to the rank of a matrix?
5. What are the conditions for a set of vectors to form a basis in higher dimensions?

**Tip:** Always check the determinant when verifying if vectors are linearly independent—it’s a quick way to determine if a set can form a basis.

Which set of vectors are linearly independent? Which set of vectors form a basis for R^2?

Understand how to determine linear independence and find the basis of a vector space using determinants. This solution involves calculating the determinants of 2x2 matrices formed by vectors to verify independence and check if they form a basis for R^2.

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