# Questions tagged [linear-independence]

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### How to prove for $AX=B$ , number of linearly independent columns in B can't exceed A? [duplicate]

I want a way to prove that for matrices $A(n \times m)$, $X(m \times n)$ and $B(n \times n)$ such that $AX=B$ and the number of linearly independent columns $k$ in $A$ with $k<n$, the maximum ...
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### Are any 2 linear combinations of a set of linearly dependent vectors linearly dependent?

I am trying to prove whether the following statement is true or not: that if we have some linearly dependent vectors, any 2 linear combinations of these vectors constitutes a linearly dependent set ...
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### Show that $F(t) = (\cos t, \sin t)$ is locally one-to-one but not globally one-to-one.

I am studying Implicit Function Theorem and Inverse Function Theorem. The problem I want to ask is: Show that $F(t) = (\cos t, \sin t)$ is locally one-to-one but not globally one-to-one. I have two ...
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### Prove that set of columns of an incidence matrix M with integral values mod 2 are linearly independent iff the corresponding set of edges are acyclic

This is a question from CLRS 3E, 16-3a. Argue that a set of columns of an incident Matrix M is linearly independent over the field of integers modulo 2 if and only if the corresponding set of edges ...
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### An analogous equivalence to one of linear independence

In a finite dimensional vector space $V$, a subset $S \subseteq V$ is said to be linearly dependent if there is a nontrivial solution $(a_i)$ to the equation  \sum_{\mathbf{v_i}\in S}a_i\mathbf{v_i} ...
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### Conditions for linear independence given the family of vectors is pairwise independent

let $n>0, n \in \mathbb{N}$ Given $n+1$ vectors $a_0, a_1, ... a_n$ which are pairwise independent i.e. for all $i,j \in \{ 0,1,...,n \}$ and $i\neq j$, $a_i$ and $a_j$ are linearly independent. ...
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### Generalisation of Stone-Weierstrass Theorem / Fourier series for linearly independent functions.

In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval $[a, b]$ can be uniformly approximated as closely as desired by a ...
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