Would this be a valid proof for linear independence?

Question

The question is a series of questions but this part gives an equation where A is a 3x3 nilpotent matrix with an index of nilpotence $k$ of 3 so that $A^k = 0$ and $A\vec{x}$ and $A^2\vec{x}$ is non-zero. I need to prove that the vectors $\vec{x}, A\vec{x}$ and $A^2\vec{x}$ so that I can extend it to a general nilpotent matrix of $n$ x $n$ with an index $k$ and show that $k\leq n$. I'm not allowed to use any theories outside of linear independence so.... $$ \lambda_1\vec{x} + \lambda_2A\vec{x} + \lambda_3A^2\vec{x}= \vec{0} \\ \lambda_1A\vec{x} + \lambda_2A^2\vec{x} + \lambda_3A^3\vec{x}= \vec{0} \\ \lambda_1A\vec{x} + \lambda_2A^2\vec{x} = \vec{0} $$ Would it be valid to show linear independence of $A\vec{x}$ and $A^2\vec{x}$ by the following method? Since $-\frac{\lambda_1}{\lambda_2}A \neq A^2$ $$ \lambda_1A\vec{x} = -\lambda_2A^2\vec{x} \\ -\frac{\lambda_1}{\lambda_2}A\vec{x}=A^2\vec{x} $$ This proof seems iffy but I have no idea how to prove it otherwise. Would appreciate any help I can get!

Answer 1

Your proof is not OK. First of all, it is all symbols, and very little text. You should always write proofs in such a way that they can be read out loud. With full sentences, and text explaining what each equation means. Most proofs include the words "let", "therefore", "implies" and the like.

Second of all, two logical errors you make are:

1. You assert that $-\frac{\lambda_1}{\lambda_2}A\neq A^2$. How do you know this is true?
2. The fact that $-\frac{\lambda_1}{\lambda_2}A\neq A^2$ is not enough to conclude, for some vector $x$, that $-\frac{\lambda_1}{\lambda_2}Ax\neq A^2x.$ It is perfectly possible for two non-equal matrices to act identically on one particular vector.

Finally, your proof is not OK because the very statement you are trying to prove is false. For example, if $A=I$, then $Ax$ and $A^2x$ will never be independent.