Question on "Intersection of subspaces is nonzero" We are asked to prove that if $V, W$ are 3-dimensional subspaces of $\mathbb{R}^5$, then $V \cap W \neq \{0\}$.
The question is answered here: https://math.stackexchange.com/a/565932/1013993
I am confused when the author writes that

So that, there exist linear combination $c_1v_1 + c_2v_2 + c_3v_3 + d_1w_1 + d_2w_2 + d_3w_3 =0$ such that coefficients $c_1,c_2,c_3,d_1,d_2,d_3$ are NOT ALL zeros.

How can we prove such a combination must exist?

Given any bases $\{v_i\}_{i=1}^3,\{w_i\}_{i=1}^3$ for $V,W$ respectively, since they in total have $6$ elements but $\mathbb{R}^5$ is of dimension $5$, then $\{v_1,v_2,v_3,w_1,w_2,w_3\}$ must be a linearly dependent set.
Hence the claim: it is from the definition of linear dependence.
