To determine whether a set of vectors is linearly independent or linearly dependent, we check whether there exists a non-trivial linear combination of the vectors that equals the zero vector. In other words, for vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n$, they are linearly dependent if there exist scalars $c_1, c_2, \dots, c_n$ (not all zero) such that:

$c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \dots + c_n \mathbf{v}_n = 0$

Alternatively, if the vectors form a matrix (with vectors as columns or rows), we check if the determinant is zero or if the system of equations has a non-trivial solution.

We will examine each case individually:

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a) $\{(1, 2, 5), (1, 3, 1), (2, 5, 7), (3, 1, 4)\}$

This is a set of 4 vectors in $\mathbb{R}^3$. The maximum number of linearly independent vectors in $\mathbb{R}^3$ is 3, so these vectors must be linearly dependent. Any set of more than 3 vectors in $\mathbb{R}^3$ is linearly dependent.

Conclusion: Linearly Dependent

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b) $\{(1, 2, 5), (2, 5, 1), (1, 5, 2)\}$

We have 3 vectors in $\mathbb{R}^3$. To determine linear dependence, we can set up the matrix and compute its determinant. If the determinant is 0, the vectors are linearly dependent. Otherwise, they are independent.

The matrix formed by these vectors is:

1 & 2 & 1 \\ 2 & 5 & 5 \\ 5 & 1 & 2 \end{pmatrix}$$ We compute the determinant: $$\text{det} = 1 \cdot \begin{vmatrix} 5 & 5 \\ 1 & 2 \end{vmatrix} - 2 \cdot \begin{vmatrix} 2 & 5 \\ 5 & 2 \end{vmatrix} + 1 \cdot \begin{vmatrix} 2 & 5 \\ 5 & 1 \end{vmatrix}$$ Calculating each minor: $$\begin{vmatrix} 5 & 5 \\ 1 & 2 \end{vmatrix} = (5)(2) - (5)(1) = 10 - 5 = 5$$ $$\begin{vmatrix} 2 & 5 \\ 5 & 2 \end{vmatrix} = (2)(2) - (5)(5) = 4 - 25 = -21$$ $$\begin{vmatrix} 2 & 5 \\ 5 & 1 \end{vmatrix} = (2)(1) - (5)(5) = 2 - 25 = -23$$ Now, substitute back: $$\text{det} = 1 \cdot 5 - 2 \cdot (-21) + 1 \cdot (-23) = 5 + 42 - 23 = 24$$ Since the determinant is **not zero**, the vectors are **linearly independent**. ### **Conclusion**: **Linearly Independent** --- ### **c)** $$\{(1, 2, 3), (1, 5, 6), (0, 0, 0)\}$$ One of the vectors is the **zero vector** $$(0, 0, 0)$$. Any set of vectors that includes the zero vector is always **linearly dependent**, since the zero vector can be written as a linear combination of any other vectors (with all coefficients being zero). ### **Conclusion**: **Linearly Dependent** --- ### **d)** $$\{(1, 2, 3), (1, 3, 5), (1, 0, 1), (2, 3, 0)\}$$ We have **4 vectors** in $$\mathbb{R}^3$$, so these vectors must be **linearly dependent** because the maximum number of linearly independent vectors in $$\mathbb{R}^3$$ is 3. ### **Conclusion**: **Linearly Dependent** --- ### Summary: a) **Linearly Dependent** b) **Linearly Independent** c) **Linearly Dependent** d) **Linearly Dependent** --- #### Would you like a more detailed explanation on how to solve any of these cases, or do you have any other questions? ### Related Questions: 1. How do we determine if a set of vectors in $$\mathbb{R}^n$$ is linearly independent or dependent? 2. What is the rank of a matrix, and how does it relate to linear dependence? 3. How do we calculate the determinant of a 3x3 matrix? 4. What is the geometric interpretation of linearly independent vectors in $$\mathbb{R}^3$$? 5. How can we determine the null space of a matrix? #### Tip: If you encounter vectors in $$\mathbb{R}^n$$, remember that the maximum number of linearly independent vectors is $$n$$. Any set of vectors with more than $$n$$ vectors in $$\mathbb{R}^n$$ is linearly dependent.