Determine if each of the following sets of vectors is linearly independent or linearly dependent.
(a) V = {[1 0 1], [1 2 1], [2 2 2]}
(b) V = {[2 1 0], [1 2 0], [3 3 1]}
(c) V = {[2 1], [1 2]}
(d) V = {[1 2 3], [4 5 6], [5 7 9]}

Let's determine whether each set of vectors is linearly independent or dependent. We'll use the determinant for square matrices or look for linear combinations in non-square matrices.

### (a) \( V = \left\{ \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix} \right\} \)

Form the matrix \( A \) with the vectors as columns:

\[
A = \begin{bmatrix}
1 & 1 & 2 \\
0 & 2 & 2 \\
1 & 1 & 2 \\
\end{bmatrix}
\]

Calculate the determinant:

\[
\text{det}(A) = 1 \cdot (2 \cdot 2 - 1 \cdot 2) - 1 \cdot (0 \cdot 2 - 1 \cdot 2) + 2 \cdot (0 \cdot 1 - 2 \cdot 1) = 1(4 - 2) - 1(0 - 2) + 2(0 - 2) = 2 + 2 - 4 = 0
\]

Since the determinant is zero, the vectors are **linearly dependent**.

### (b) \( V = \left\{ \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 1 \end{bmatrix} \right\} \)

Form the matrix \( A \):

\[
A = \begin{bmatrix}
2 & 1 & 3 \\
1 & 2 & 3 \\
0 & 0 & 1 \\
\end{bmatrix}
\]

Calculate the determinant:

\[
\text{det}(A) = 2 \cdot (2 \cdot 1 - 3 \cdot 0) - 1 \cdot (1 \cdot 1 - 3 \cdot 0) + 3 \cdot (1 \cdot 0 - 2 \cdot 0) = 2(2) - 1(1) + 3(0) = 4 - 1 = 3
\]

Since the determinant is non-zero, the vectors are **linearly independent**.

### (c) \( V = \left\{ \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \end{bmatrix} \right\} \)

Form the matrix \( A \):

\[
A = \begin{bmatrix}
2 & 1 \\
1 & 2 \\
\end{bmatrix}
\]

Calculate the determinant:

\[
\text{det}(A) = 2 \cdot 2 - 1 \cdot 1 = 4 - 1 = 3
\]

Since the determinant is non-zero, the vectors are **linearly independent**.

### (d) \( V = \left\{ \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}, \begin{bmatrix} 5 \\ 7 \\ 9 \end{bmatrix} \right\} \)

Form the matrix \( A \):

\[
A = \begin{bmatrix}
1 & 4 & 5 \\
2 & 5 & 7 \\
3 & 6 & 9 \\
\end{bmatrix}
\]

Calculate the determinant:

\[
\text{det}(A) = 1 \cdot (5 \cdot 9 - 7 \cdot 6) - 4 \cdot (2 \cdot 9 - 7 \cdot 3) + 5 \cdot (2 \cdot 6 - 5 \cdot 3)
\]

\[
\text{det}(A) = 1(45 - 42) - 4(18 - 21) + 5(12 - 15) = 1(3) - 4(-3) + 5(-3) = 3 + 12 - 15 = 0
\]

Since the determinant is zero, the vectors are **linearly dependent**.

### Summary:
- (a) Linearly Dependent
- (b) Linearly Independent
- (c) Linearly Independent
- (d) Linearly Dependent

Would you like further details or a deeper explanation?

Here are some related questions:
1. How can you determine linear dependence using the rank of a matrix?
2. What is the significance of linear independence in vector spaces?
3. How does the concept of linear dependence apply to solving systems of equations?
4. What geometric interpretation can be made about linearly dependent vectors?
5. How can you determine if a larger set of vectors is linearly dependent? 

**Tip:** When checking for linear dependence in larger sets, if any vector can be written as a linear combination of others, the

Learn how to determine if sets of vectors are linearly dependent or independent through matrix determinant calculations. This detailed guide covers examples (a), (b), (c), and (d) with step-by-step solutions. Suitable for college-level students studying linear algebra.

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