$A$ be $n×n$ matrix $A^{n}=0$ ,$A^{n-1}$ not equal to zero a vector $v$ belongs to R^n.then how to proof {V,AV,...A^(n-1)V} is a basis.
115273
Nilpotent matrix and basis for $F^n$
146243
Does there exist a matrix $\mathbf{A}\in\mathbb{R}^{3\times3}$ such that $\mathbf{A}^{2}=-\mathbf{I}$?
18297
Suppose that each of the row sums of an $n\times n$ matrix $A$ is equal to zero. Show that $A$ must be singular.
150720
Given that $W$ is a subspace of $V$, prove: $\dim W=\dim V \implies W=V$
132879
Matlab questions about $a_{n+1} = 4a_{n} + n +1$
55752
Prove that $n+1$ vectors in $\mathbb{R}^n$ cannot be linearly independent
44596
If $A$ is a square matrix and $A^2 = 0$ then $A=0$. Is this true? If not, provide a counter-example.
121982
Show matrix A is invertible if and only if $a_0 \neq 0$ with $A^n+a_{n-1}A^{n-1}+....+a_1A+a_0I=O$.
8291
If $n$ vectors are linearly independent, is their span $\mathbb{R}^n$?
101506
Why is $1, (x-5)^2, (x-5)^3$ a basis of $U=\{p \in \mathcal P_3(\mathbb R) \mid p'(5)=0\}$?
76498
Does every linearly independent set of n vectors in $R^n$ forms a basis in $R^n$?
81879
$Av=λv$ basis proof problems
69980
Proving that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$
104653
$A$ and $B$ are $3\times 3$ real matrices such that $\operatorname{rank}(AB)=1$, then $\operatorname{rank}(BA$) can not be which of the following?
58021
Finding matrix $B$ is not zero matrix where $AB= 0$
57771
Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$.
61274
If $AA^T$ is the zero matrix, then $A$ is the zero matrix