Let be linear transformations on an -dimensional vector space such that and for . Then .

How does this happen? Take the expression and multiply by any on both sides. We see that . Hence any vector can be expressed as a sum of elements in for .

Why do we have a direct sum decomposition? Let for . Then consider . For any where , $v_i=E_i v$ for some . Hence . Hence, we have . Now for some . Hence . Note that (just multiply the expression by on both sides). Hence, we have . Now . Hence, we have . This is true for all . Hence, all the , which proves that we have a direct sum decomposition of .

Why is all this relevant? Because using the minimal polynomial of any transformation , we can construct such ‘s which satisfy the above two conditions, and can hence decompose the vector space as a direct sum of subspaces. Moreover, these subspaces have the additional property that they’re -invariant. Each