Linear dependence of functionals (Intuition)
Solution 1:
It's kind of natural to see when $X$ is a Hilbert space. Each $\varphi_j$ is of the form $$\varphi_j(x)=\langle x,y_j\rangle,\qquad \varphi(x)=\langle x,y\rangle. $$ So $$\ker\varphi_j=\{y_j\}^\perp,\qquad \ker\varphi=\{y\}^\perp.$$ Then the condition becomes $$\{y\}^\perp\supset\bigcap_j\{y_j\}^\perp=(\operatorname{span}\{y_1,\ldots,y_n\})^\perp.$$ Taking orthogonals, $$\{y\}\subset\operatorname{span}\{y_1,\ldots,y_n\}.$$