Let $V$ be a $K$-vector space, $f: V \rightarrow V$ a linear map. Under some condition, show that $v, f(v),...,f^n(v)$ are linearly independent.
132492
If a linear transformation $T$ has $z^n$ as the minimal polynomial, there is a vector $v$ such that $v, Tv,\dots, T^{n-1}v$ are linearly independent
28289
Show that $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$ is linear and continuous
48990
Linear map $f:V\rightarrow V$ injective $\Longleftrightarrow$ surjective
96354
Linear independence of $\sin(x)$, $\sin(2x)$, $\sin(3x)$ in Map($\mathbb{R},\mathbb{R}$)
92291
$U_1, U_2, ..., U_k$ subspaces of linear space $X$ with same dimensions, prove $X=U_1\oplus V=U_2\oplus V=...=U_k\oplus V$
133087
Prove that, if $\{u,v,w\}$ is a basis for a vector space $V$, then so is $\{u+v, v+w, u+v+w\}$.
52596
A basis for $k(X)$ regarded as a vector space over $k$
90691
Linear independence of vectors over larger fields
54737
Proof that $V = \sum_k U_k$ and $\dim V=\sum_k \dim U_k$ implies $V=\oplus_k U_k$
66283
Let V be a vector space. If every subspace of V is T-invariant, prove that there exist a scalar multiple c such that T=c1v
140095
Let $k^p = k$, $K = k(x)$ and $F = k(u)$ for some $u \in K$. Show that $K/F$ is separable $\iff u \not\in K^p$.
95928
Independence and orthogonality
113671
Frobenius Automorphism as a linear map
127000
When does $V = \ker(f) \oplus \operatorname{im}(f)$?
107248
Prove that the set $ \{\sin(x),\cos(x),\sin(2x),\cos(2x)\}$ is linearly independent.
102529
Prove that the set $\left\{ x, Ax, \dots, A^{k-1} x \right\}$ is linearly independent
53528
If $\ker f\subset \ker g$ where $f,g $ are non-zero linear functionals then show that $f=cg$ for some $c\in F$.