To find a value of $t$ such that the vectors $(3, 1, 4)$, $(2, -3, 5)$, and $(5, 9, t)$ are not linearly independent in $\mathbb{R}^3$, we need to determine when these vectors become linearly dependent.

Step-by-step solution:

1. Linear dependence condition: Vectors are linearly dependent if there exist constants $c_1, c_2, c_3$, not all zero, such that: $c_1 \cdot (3, 1, 4) + c_2 \cdot (2, -3, 5) + c_3 \cdot (5, 9, t) = (0, 0, 0)$ In matrix form, this becomes:

   3 & 2 & 5 \\ 1 & -3 & 9 \\ 4 & 5 & t \end{pmatrix}$$ We are looking for values of $$t$$ such that the determinant of this matrix is zero, because the determinant of the coefficient matrix being zero implies linear dependence.

2. Determinant calculation: The determinant of the matrix is given by:

   3 & 2 & 5 \\ 1 & -3 & 9 \\ 4 & 5 & t \end{pmatrix} = 3 \cdot \det \begin{pmatrix} -3 & 9 \\ 5 & t \end{pmatrix} - 2 \cdot \det \begin{pmatrix} 1 & 9 \\ 4 & t \end{pmatrix} + 5 \cdot \det \begin{pmatrix} 1 & -3 \\ 4 & 5 \end{pmatrix}$$ Let's compute these 2x2 determinants step by step. - $$\det \begin{pmatrix} -3 & 9 \\ 5 & t \end{pmatrix} = (-3)(t) - (9)(5) = -3t - 45$$ - $$\det \begin{pmatrix} 1 & 9 \\ 4 & t \end{pmatrix} = (1)(t) - (9)(4) = t - 36$$ - $$\det \begin{pmatrix} 1 & -3 \\ 4 & 5 \end{pmatrix} = (1)(5) - (-3)(4) = 5 + 12 = 17$$

3. Substitute the results into the determinant expression: $\det = 3 \cdot (-3t - 45) - 2 \cdot (t - 36) + 5 \cdot 17$ Simplifying each term: $= 3(-3t - 45) = -9t - 135$ $= -2(t - 36) = -2t + 72$ $= 5 \cdot 17 = 85$ Now combine all the terms: $\det = (-9t - 135) + (-2t + 72) + 85$ Simplify further: $\det = -9t - 135 - 2t + 72 + 85 = -11t + 22$

4. Set the determinant equal to zero: For the vectors to be linearly dependent, the determinant must be zero: $-11t + 22 = 0$ Solving for $t$: $t = \frac{22}{11} = 2$

Conclusion:

The vectors $(3, 1, 4)$, $(2, -3, 5)$, and $(5, 9, t)$ are linearly dependent when $t = 2$.

Would you like any more details on this, or do you have further questions?

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Here are 5 follow-up questions:

1. What is the geometrical interpretation of linear dependence in $\mathbb{R}^3$?
2. How can we determine if more than three vectors are linearly dependent in $\mathbb{R}^3$?
3. What happens to the determinant when the vectors are linearly independent?
4. How do you interpret the determinant of a 2x2 matrix geometrically?
5. What is the relationship between the rank of a matrix and the linear dependence of its rows or columns?

Tip: In $\mathbb{R}^3$, if three vectors are linearly dependent, they all lie in the same plane.