I'm learning about Basis and Spans and now that's I've figured out what these are, I'm trying to understand the Replacement Theorem(also called the Exchange Theorem).
The definition goes like this:
Let $V$ be a vector space and let $B = \{v_{1}, v_{2}, …, v_{n}\}$ be a basis for V with n elements. Choose and integer $m\leq n$ and let $S = \{w_1, w_2, …, w_m\}$ be a finite set of linearly independent vectors. Then, there is a set $Z\subset B$ containing exactly $n-m$ elements such that $span({Z\cup S}) = V$
Now this is what I understand from the above definition. $V$ is a vector space and $B$ being the basis of $V$ is the set of linearly independent vectors such that for any vector, for example, $u_1 \in V$, we have:
$u_1 = \sum u_1v_i$ where $i$ is the number of elements in $B$. If, now, I make a set $S$ with $n-m$ or $n=m$ elements then I'll have the remaining vectors $w_i$ in my set $Z$. These vectors are also linearly independent.
Finally, set $span(\{Z \cup S\}) = V$. Now that's my problem:
Why it is not $span(\{Z\cup S\}) = B$ since $S$ is a set of linearly independent vectors(which is a basis) and if I'm making a set $Z$ with $n-m$ elements(which is from $B$ as well) ?
It would really help if an example is given in the explanation, let's say we have a vector space $V = \{u_1, u_2, u_3, u_4\}$.
Thanks
Best Answer
Perhaps I've misunderstood the issue but there seems to be some confusion about the difference between a basis, a vector space, and the span operation. A basis is a set of $n$ vectors. The vector space equals the set of all linear combinations of elements of a basis. The span of a set of vectors is the linear combination of all the elements in that set.
$$span(B) = \{\sum_i c_i v_i\}$$ where $c_i$ are arbitrary scalars.
You would not say that the span of a set of vectors is equal to an $n$ point set as in $span(Z \cup S) = B$. Instead you might ask if it equals the entire vector space $V$ or a subspace.
So for example. The vector space $\mathbb{R}^2$ has a basis $v_1, v_2$ which could be the $(1, 0)$ and $(0, 1)$ vectors. The elements of that space are vectors $(c_1, c_2)$ where $c_1, c_2$ are real numbers. The basis is a two element set $\{v_1, v_2\}$. The span of, for example, the one element set $\{v_1\}$ is all the vectors of the form $(c_1, 0)$ where $c_1$ is a real number. That is an infinite set.