--- Commentary: $\textbf{Exercise 4.}$ (a) Show that a list of length one in a vector space is linearly independent if and only if the vector in the list is not 0. (b) Show that a list of length two in a vector space is linearly independent if and only if neither of the two vectors in the list is a scalar multiple of the other. $\textbf{Solution 4.}$ (a) Suppose $v \in V$. If $v = 0$, then the list $v$ of length one is not linearly independent because $1v = 0$. Conversely, if $v \neq 0$, then the list $v$ of length one is linearly independent because the only scalar $a \in \mathbb{F}$ such that $av = 0$ is $a = 0$. (b) Suppose $v_1, v_2 \in V$. Suppose there is a scalar $a \in \mathbb{F}$ such that $v_1 = av_2$ or $v_2 = av_1$. Then $1v_1 - av_2 = 0$ or $av_1 - 1v_2 = 0$. Thus the list $v_1, v_2$ of length two is linearly dependent. Conversely, suppose the list $v_1, v_2$ of length two is linearly dependent. Then there exist scalars $a_1, a_2 \in \mathbb{F}$, not both 0, such that $a_1v_1 + a_2v_2 = 0$. If $a_1 \neq 0$ then $v_1 = -\frac{a_2}{a_1}v_2$. If $a_2 \neq 0$ then $v_2 = -\frac{a_1}{a_2}v_1$. $\textit{Commentary:}$ This exercise explores the basic conditions for linear independence in short lists of vectors. Part (a) shows that a single vector is linearly independent if and only if it's not the zero vector. This is because the only way a single vector can be linearly dependent is if it's a scalar multiple of the zero vector, which is only possible if the vector is itself zero. Part (b) shows that two vectors are linearly independent if and only if neither is a scalar multiple of the other. If one vector is a scalar multiple of the other, then there's a non-trivial linear combination of the vectors that equals zero, making the list linearly dependent. Conversely, if the list is linearly dependent, then there's a non-trivial linear combination of the vectors that equals zero, which implies that one vector must be a scalar multiple of the other. These conditions are fundamental in understanding linear independence and are often used as the base cases in proofs involving linear independence of longer lists of vectors. $\textit{Additional Examples:}$ In $\mathbb{R}^2$, the list $(1, 2)$ is linearly independent, but the list $(0, 0)$ is not. In $\mathbb{C}^3$, the list $(1, i, 0)$, $(2, 2i, 0)$ is linearly dependent, because $(2, 2i, 0) = 2(1, i, 0)$. In $\mathbb{F}_5^2$, the list $(1, 2)$, $(3, 1)$ is linearly independent, because neither vector is a scalar multiple of the other (in $\mathbb{F}_5$). In the space of polynomials $\mathbb{Q}[x]$, the list $1$, $x$ is linearly independent, but the list $x$, $2x$ is linearly dependent. In the space of continuous functions $C([0,1], \mathbb{R})$, the list $e^x$ is linearly independent, but the list $\sin(x)$, $2\sin(x)$ is linearly dependent. These examples illustrate the basic conditions for linear independence in various contexts, including - over the real and complex numbers, - over a finite field, - in a polynomial space, and - in a function space. - The conditions are the same in each case: a single vector is linearly independent unless it's zero, and two vectors are linearly independent unless one is a scalar multiple of the other.