suppose n-by-n matrix A only has zero eigenvalues, and $A^{n-1} \neq 0$ show there exists x such that x, Ax, ...$A^{n-1}$x are linearly independent.
109228
Prove that $v, Tv, T^2v, ... , T^{m-1}v$ is linearly independent
43282
Why do $n$ linearly independent vectors span $\mathbb{R}^{n}$?
123803
If $X^n$ is a diagonal matrix with distinct eigenvalues, then is $X$ also a diagonal matrix with distinct eigenvalues?
14320
How to prove that eigenvectors from different eigenvalues are linearly independent
994
Linear independence of vectors that form non-square matrix
86125
If $A$ is idempotent, then the eigenvalues of $A$ are $0$ or $1$
60723
Prove that if $A$ is normal, then eigenvectors corresponding to distinct eigenvalues are necessarily orthogonal (alternative proof)
157269
If $\alpha_{1},\ldots,\alpha_{m}$ are vectors different from zero vector, then there is a linear functional $f$ such that $f(\alpha_{i})\neq 0$
8291
If $n$ vectors are linearly independent, is their span $\mathbb{R}^n$?
138343
How to prove that the set $\{\sin(x),\sin(2x),...,\sin(mx)\}$ is linearly independent?
19649
A question about the proof of "a symmetric matrix has real eigenvalues".
30017
Orthogonality and linear independence
822
Given that $X,Y$ are independent $N(0,1)$ , show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}}$ are independent $N(0,\frac{1}{4})$
158724
Show that $B$ is diagonalizable if If $AB=BA$ and $A$ has distinct real eigenvalues
56102
Prove that there is no non-zero linear operator on $\mathbb C^{2}$ such that $(\alpha | T\alpha) = 0$
27248
If $A$ is an $m \times n$ matrix, prove that $Ax=0$ has infinite solutions $\iff$ $\text{rank}(A)<n$
30720
If the dot product between two vectors is $0$, are the two linearly independent?