Consider the matrix $$A=\left(\begin{array}{rrr}
1 & -1 & 5 \\
1 & -1 & 4 \\
1 & -1 & 3
\end{array}\right)$$
We can see that $$1\vec{C_1}+1\vec{C_2}+0\vec{C_3}=\vec{0}$$
and we say that that the three column vectors $\vec{C_1},\vec{C_2},\vec{C_3}$ are Linearly dependent. But we cannot express $\vec{C_3}$ in terms of $\vec{C_1}, \vec{C_2}$ right?
Because i read the definition as:
The set of vectors $\left\{x_{1}, x_{2}, \ldots, x_{k}\right\}$ is linearly dependent if
$$
r_{1} x_{1}+r_{2} x_{2}+\cdots+r_{k} x_{k}=0
$$
for some $r_{1}, r_{2}, \ldots, r_{k} \in \mathbb{R}$ where at least one of $r_{1}, r_{2}, \ldots, r_{k}$ is non-zero.
Best Answer
Linearly dependent sets don't require that EVERY vector be able to be written as a sum of the others, just that at least ONE can be. In this case, you can write the first two in terms of each other.