2
$\begingroup$

Assume the Exchange Theorem and prove the following: Assume the vector space V is finitely generated. Then there is a natural number n such that the length of a linearly independent sequence is less than or equal to n.

Theorem: Assume V is spanned by a sequence $(v_1, . . . , v_n)$ of length n. Then any sequence $(w_1, . . . , w_{n+1})$ of length $n + 1$ is linearly dependent.

Approach

Proof

Let x be the cardinality of a linearly independent sequence

Assume a vector space V is finitely generated by a spanning sequence $B=(v_1,...,v_k)$. By the exchange theorem, if we append a $v_j \in V\B$, $B$ becomes linearly dependent, so there doesn't exist a linearly independent sequence of cardinality greater than $k$.

If $B$ is linearly independent $|B|=k=x$.

If $B$ is linearly dependent then,there exists maximal linearly independent spanning sub-sequence $(v_1,..,v_j)$ for $j<k$,so $j=x$

How does that look?

$\endgroup$
1
  • 1
    $\begingroup$ Doesn't the exchange theorem tell you exactly that the cardinality of a spanning set is an upper bound for the cardinality of a linearly independent set? What is the statement you call “Exchange theorem”? $\endgroup$
    – egreg
    May 23, 2016 at 20:17

1 Answer 1

1
$\begingroup$

I would not go such a way because what does it mean to have V-B. One is a space the other is a sequence. You can do V- span (B). Depending on how formal you want to be you can make many arguments.

Proof by contradiction assuming the $\{w_1,\dots,w_{n+1}\}$ is Linearly independent. Say dim $V= n$ because there are $n$ elements in the basis. Now $w_i \in V$ for all i. Now $span (\{w_1,\dots,w_{n+1}\})$ is a subspace of V. Well if you are a subspace of a space $dim V \geq dim W$ where W is any subspace of V. Now if let $W= span (\{w_1,\dots,w_{n+1}\})$. Well because this list is linearly independent they form a basis for W ie dim $W=n+1$.

This is a contradiction $dim V \geq dim W$, but you get $n \geq n+1$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .