--- Commentary: $\textbf{Exercise 5.}$ Find a number $t$ such that $(3, 1, 4), (2, -3, 5), (5, 9, t)$ is not linearly independent in $\mathbb{R}^3$. $\textbf{Solution 5.}$ We begin by looking just at the first two coordinates of each vector above. To write $(5, 9)$ as a linear combination of $(3, 1)$, $(2, -3)$, we must find $a, b \in \mathbb{R}$ such that $a(3, 1) + b(2, -3) = (5, 9),$ which is equivalent to the system of equations \begin{align*} 3a + 2b &= 5 \ a - 3b &= 9. \end{align*} Solving for $a, b$, we get $a = 3$, $b = -2$. Thus to choose $t$ so that $(3, 1, 4)$, $(2, -3, 5)$, $(5, 9, t)$ is linearly dependent, we need $3(3, 1, 4) - 2(2, -3, 5) = (5, 9, t),$ which implies that $t = 2$. $\textit{Commentary:}$ This exercise asks to find a value of $t$ that makes a list of three vectors in $\mathbb{R}^3$ linearly dependent. The key observation is that if the list is linearly dependent, then one of the vectors must be a linear combination of the other two. The solution begins by ignoring the third coordinate and finding the unique linear combination of the first two vectors that produces the first two coordinates of the third vector. This is done by setting up and solving a system of linear equations. Once the coefficients of this linear combination are found, they are used to determine the value of $t$ that would make the linear combination equal to the third vector, making the list linearly dependent. This exercise demonstrates the connection between linear dependence and the ability to express one vector as a linear combination of others. It also provides practice in setting up and solving systems of linear equations. $\textit{Additional Examples:}$ In $\mathbb{R}^4$, the list $(1, 2, 3, 4)$, $(0, 1, 1, 1)$, $(2, 5, t, 9)$ is linearly dependent if and only if $t = 7$. In $\mathbb{C}^3$, the list $(1, i, -1)$, $(i, -1, i)$, $(2+i, t, -2+i)$ is linearly dependent if and only if $t = -1-i$. In $\mathbb{Q}^3$, the list $(\frac{1}{2}, 1, \frac{3}{2})$, $(1, -1, 2)$, $(\frac{3}{2}, t, \frac{7}{2})$ is linearly dependent if and only if $t = 0$. In the space of polynomials $\mathbb{R}[x]$, the list $1+x$, $1-x$, $t+x$ is linearly dependent if and only if $t = 2$. In the space of continuous functions $C([-1,1], \mathbb{R})$, the list $e^x$, $\sin(x)$, $te^x + \cos(x)$ is linearly dependent if and only if $t = 1$. These examples illustrate the process of finding values that make a list of vectors linearly dependent in various contexts, including over the real and complex numbers, over the rational numbers, in a polynomial space, and in a function space. In each case, the value is found by expressing the last vector as a linear combination of the others and solving for the unknown coefficient.