利用dim(W1+W2)>=max{dim(W1), dim(W2)} >=min{dim(W1), dim(W2)} >=dim(W1∩W2)=dim(W1+W2)-1 dim(W1)+dim(W2)=dim(W1+W2)+dim(W1∩W2)=2dim(W1∩W2)=,dim(W1+W2)-1=dim(W1∩W2), ===》 ...

• max{dim(W1), dim(W2)} >=min{dim(W1), dim(W2)}

• dim(W1+W2)>dim(W1∩W2)=dim(W1+W2)-1

• 2dim(W1∩W2)+2=2dim(W1+W2)>=dim(W1)+dim(W2)=dim(W1+W2)+dim(W1∩W2)=2dim(W1∩W2)+1

===》dim(W1+W2)=max{dim(W1), dim(W2)}
min{dim(W1), dim(W2)}=dim(W1∩W2)=dim(W1+W2)-1

然后根据与一个有限维线性空间V的维数相等的子空间是线性空间本身

证明