Prove matrix is linear independence given $M^n=0$

Question

Denote matrix $M \in M_{n×n}$($\mathbb{R}$) $st$ $M^n=0$ and $n \in $$\mathbb{N}$. Prove that I, M, · · · , $M^{n−1}$ are linearly independent $\iff$ $M^{n−1} \neq 0$. Started by assuming linear independence as $r_1×I+r_2×M...+r_{n-1}×M^{n-1}=0$ and prove the consequent. Tried a proof by contradiction: assume that $M^{n−1}$ = $0$. Then the sum would not be linearly independent. Is this on the right track? No idea where to go from here. More help with the opposite direction also needed.

Answer 1

The forward direction is quite straightforward. Let us prove the contrapositive of this statement. If $M^{n-1}$ were zero, then the nontrivial linear combination composed of only $M^{n-1}$ yields zero, so the elements cannot be independent.

For the backward direction, we will also prove the contrapositive: if $I, M, \cdots , M^{n-1}$ are dependent, then $M^{n-1} = 0$. Suppose there is a nontrivial linear combination of these elements yielding zero, and let $M^k$ be the smallest power of $M$ that has a nonzero coefficient in this linear combination. Multiplying on both sides of the linear combination by $M^{n - k - 1}$, we obtain the equation $M^{n-1} = 0$ as desired, since by assumption all terms with lower powers of $M$ are zero and all terms with higher powers have power at least $n$ and are hence annihilated.

Answer 2

Hint: Proving $M^{n-1} = 0 \implies$ the set $\{I,M,\dots,M^{n-1}\}$ is linearly dependent (not linearly independent) is easy: simply note that any set that includes the zero-vector (or in this case the zero matrix, since our vector space is that of the set of matrices) must be linearly dependent.

The inverse implication is trickier. Suppose that $M^{n-1} \neq 0$, and suppose that $c_0,c_1,\dots,c_{n-1}$ are such that $$ c_0 I + c_1 M + \cdots + c_{n-2} M^{n-2} + c_{n-1} M^{n-1} = 0. $$ We want to show that the coefficients $c_0,\dots,c_{n-1}$ must all be zero, showing that the set is linearly independent. Multiply both sides of the above equation by $M^{n-2}$ and simplify; what does this tell you about the coefficients?