A new proof of Stanley's theorem on the strong Lefschetz property

A standard graded artinian monomial complete intersection algebra $A=\mathbb{k}[x_1,x_2,\dots ,x_n]\mathrm{/}(x_1^{a_1},x_2^{a_2},\dots ,x_n^{a_n})$, with $\mathbb{k}$ a field of characteristic zero, has the strong Lefschetz property due to Stanley in 1980. In this paper, we give a new proof for this result by using only the basic properties of linear algebra. Furthermore, our proof is still true in the case where the characteristic of $\mathbb{k}$ is greater than the socle degree of $A$, namely $a_1+a_2+\cdots +a_n-n$.