[tex]\mathbf c[/tex] lies in [tex]\mathrm{span}\{\mathbf a,\mathbf b\}[/tex] if [tex]\mathbf c[/tex] can be written as a linear combination of [tex]\mathbf a[/tex] and [tex]\mathbf b[/tex].
In other words, [tex]\mathbf c\in\mathrm{span}\{\mathbf a,\mathbf b\}[/tex] if there exists scalars [tex]k_1,k_2\in\mathbb R[/tex] such that
[tex]\mathbf c=k_1\mathbf a+k_2\mathbf b[/tex]
Vectors in [tex]\mathbb R^n[/tex] are equal if their components are equal:
[tex]\langle4,1,0\rangle=k_1\langle1,3,4\rangle+k_2\langle4,9,9\rangle[/tex]
[tex]\langle4,1,0\rangle=\langle k_1+4k_2,3k_1+9k_2,4k_1+9k_2\rangle[/tex]
[tex]\implies\begin{cases}k_1+4k_2=4\\3k_1+9k_2=1\\4k_1+9k_2=0\end{cases}[/tex]
This system has no solutions, so [tex]\mathbf c[/tex] does not belong to [tex]\mathrm{span}\{\mathbf a,\mathbf b\}[/tex].
