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]}

To determine whether each set of vectors is linearly independent or linearly dependent, we will use the concept that a set of vectors is linearly independent if the only solution to the equation \(c_1\mathbf{v_1} + c_2\mathbf{v_2} + \dots + c_n\mathbf{v_n} = 0\) is \(c_1 = c_2 = \dots = c_n = 0\). This is equivalent to checking if the determinant of the matrix formed by the vectors is non-zero (for square matrices) or by checking the rank (for non-square matrices).

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

To determine if these vectors are linearly independent or dependent, we form the matrix with these vectors as columns:

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

Now, compute the determinant of this matrix:

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

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:

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

Compute 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:

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

Compute 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:

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

Compute 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 any further details or visualizations for these results?

Here are some related questions:
1. How do you find the rank of a matrix and how does it relate to linear independence?
2. What does it mean geometrically when vectors are linearly dependent in 2D or 3D space?
3. Can you always use determinants to check for linear independence in non-square matrices?
4. How do you interpret a determinant of zero in terms of the span of vectors?
5. What are the applications of linear independence in solving systems of linear equations?

**Tip:** When checking for linear dependence, if a vector is a linear combination of others, the set is dependent, even if the determinant method isn't directly applicable (e.g., for more vectors than dimensions).

Learn how to determine if sets of vectors are linearly independent or dependent using determinant calculations. Detailed solutions for sets (a) to (d) are provided, including matrix formation and determinant computation. Suitable for college-level students studying linear algebra.

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