Let $V$ be a vector space over a field $K$ and let $A = \{c_{1} , c_{2} , ...., c_{n}\}$ be a basis for $V$. If $m > n$ and $a_{1} , a_{2} , a_{3} ...., a_{m}$ are elements of $V$ , how to prove that these elements are linearly dependent?
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Steitnitz's theorem says: is you have a basis $c_1, \dots , c_n$ of a vector space $V$ and $a_1, \dots , a_m$ are linearly independent vectors, then you can substitute $m$ vectors of the basis $c_1, \dots , c_n$ by $a_1, \dots , a_m$ getting a new basis. In particular, $m\leq n$. Since you say that $m> n$, then $a_1 , \dots , a_m$ must be linearly dependent. |
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Assume that you have a linear combination of the $a_n$'s. By expressing everything up in coordinates with respect to the basis, you get a homogenous system of $n$ linear equations in $m$ unknowns (the unknowns being the coefficients of the linear combination). If this system has a nonzero solution, then the $a$'s are linearly dependent ... |
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