Show that for a set of vectors to be linearly independent in a subfield $F$ of a larger field $G$, they must be linearly independent in $G$.
90691
Linear independence of vectors over larger fields
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
72391
linear dependence proof using subsets
64395
If $v_1,...,v_r$ are eigenvectors that correspond to distinct eigenvalues, then they are linearly independent.
142229
Prove that if $V \cap \text{Span}\{u_1, u_2,...,u_k\} = \{ 0\} $, then $Au_1, Au_2,...,Au_k$ are linearly independent
144557
Linearly independent functionals
16673
If $X$ and $Y$ are independent then $f(X)$ and $g(Y)$ are also independent.
92534
Prove that any finite subset of a linearly independent set is linearly independent
40609
Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent
96354
Linear independence of $\sin(x)$, $\sin(2x)$, $\sin(3x)$ in Map($\mathbb{R},\mathbb{R}$)
92600
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.
95771
Rows linearly independent implies columns linearly independent
994
Linear independence of vectors that form non-square matrix
76498
Does every linearly independent set of n vectors in $R^n$ forms a basis in $R^n$?
59803
Converse of Schur's Lemma in finite dimensional vector spaces
27478
Proof that two basis of a vector space have the same cardinality in the infinite-dimensional case
126769
Linear independence of columns implies non-zero determinant