Say that I have a vector space $V$ and vectors $v_1, v_2, v_3$ which are linearly dependent. The definition of linear dependence is that I can find coefficients $a_1, a_2, a_3 \in F$, not all zero, so that
$$
a_1 v_1 + a_2 v_2 + a_3 v_3 = 0.
$$
Is this definition equivalent to the assertion that I can write any of these $v_i$ in terms of the other two?
If the $a_i$ is non-zero, this is straightforward to me. Say $a_1 \neq 0$. Then I subtract $a_1 v_1$ from both sides, divide by $a_1$, and rename the coefficients. In particular, I'd have
$$
v_1 = – \frac{a_2}{a_1} v_1 – \frac{a_3}{a_1} v_3,
$$
and I can define $b_1 = – \frac{a_2}{a_1}$ and $b_2 = – \frac{a_3}{a_1}$. Then I have
$$
v_1 = b_1 v_1 + b_2 v_3.
$$
If $a_1 \neq 0$, I'm not convinced that I can do this.
Let's say, for example, that $v_1$ and $v_2$ are linearly independent, but upon adding $v_3$, I get a linearly dependent set. The only non-zero coefficient would have to be $v_3$, so I should only be able to write $v_3$ in terms of $v_1$ and $v_2$.
Is there a more general statement of the equivalence?
Best Answer
Your idea does not work. To visualize an example, consider three vectors in the $x$-$y$ plane (three vectors in a two-dimensional space necessarily constitute a linearly dependent set), and then add a fourth vector parallel to the $z$-axis. There is no way to express the fourth vector as a linear combination of the first three, but the set of four vectors is linearly dependent. In fact, the only time you can express $v_1$ as a linear combination of the remaining vectors of your set is precisely when there is a linear combination summing to $0$ in which $a_1 \neq 0$.
You're asking whether $v_1 \in \operatorname{span} \{v_2, \ldots, v_n \}$. If it is, then you can express $v_1$ as a linear combination of the remaining vectors. Otherwise, you can't. In fact, that's the definition of span. Your idea works, however, if you prove that $v_1$ is an element of a minimal linearly dependent set (i.e., no proper subset is linearly dependent).
By the way, you should get in the habit of thinking about sets of vectors, rather than individual vectors, being linearly dependent or linearly independent. It's technically correct and will probably help keep you from going astray as you think about these issues.