Help with exercise 6 please? **Exercise 6**Is the set of vectors \[ V = \{t^2 + 2t - 1, 3t + 5, -t^2 + t + 6\} \]linearly dependent?**Example 6**What about this set of vectors?\[ O = \{t^2 + 2t + 1, t^2 - t + 3, 3t + 1\} \]Is it linearly dependent? Let's check. First, set it as a linear combination of the vectors set equal to the zero vector, \( \mathbf{0} \).\[ a(t^2 + 2t + 1) + b(t^2 - t + 3) + c(3t + 1) = 0 \]\[ (a + b)t^2 + (2a - b + 3c)t + (a + 3b + c) = 0 \]Again, make sure you see how this works algebraically. This yields the following system of equations, augmented matrix, and resulting RREF matrix.\[\begin{align*}a + b &= 0 \\2a - b + 3c &= 0 \\a + 3b + c &= 0\end{align*}\]\[\begin{bmatrix}1 & 1 & 0 & : & 0 \\2 & -1 & 3 & : & 0 \\1 & 3 & 1 & : & 0\end{bmatrix}\quad \overset{RREF}{\longrightarrow} \quad\begin{bmatrix}1 & 0 & 0 & : & 0 \\0 & 1 & 0 & : & 0 \\0 & 0 & 1 & : & 0\end{bmatrix}\]Remember from previous tutorials that this means that this system only has one solution. That solution is \((a, b, c) = (0, 0, 0)\). This is the trivial solution! Because this is the only solution, we do **NOT** say that the set of vectors \( O \) is linearly dependent. In fact, we say that the set of vectors \( O \) is linearly independent. Going back to

Name: Help with exercise 6 please? **Exercise 6**Is the set of vectors \[ V
Uploaded: 2024-03-20T07:07:37.000Z
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Description: Help with exercise 6 please? **Exercise 6**Is the set of vectors \[ V = \{t^2 + 2t - 1, 3t + 5, -t^2 + t + 6\} \]linearly dependent?**Example 6**What about this set of vectors?\[ O = \{t^2 + 2t + 1, t^2 - t + 3, 3t + 1\} \]Is it linearly dependent? L