首先用定义证明im(f)与ker(f)正交. 任意x∈im(f), y∈ker(f). 即有f(y) = 0, 且存在z∈V使x = f(z). 由f是对称变换, 内积(x,y) = (x,f(z)) = (f(x),z) =(0,z) = 0, 即x,y正交. 再由im(f)与ker(f)维数互补, 即知im(f)是ker(f)的正交补.。
