Math Problem Statement
Which set of vectors are linearly independent? Which set of vectors form a basis for R^2?
Solution
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 and in are linearly independent if the determinant of the matrix formed by these vectors is non-zero. The matrix can be written as:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Independence
Determinant
Vector Spaces
Basis
Formulas
det(A) = v11 * v22 - v12 * v21
Theorems
A set of vectors is linearly independent if the determinant is non-zero.
A set forms a basis if it is linearly independent and spans the space.
Suitable Grade Level
Grades 10-12