I've been reading a bit around MSE and I've stumbled upon some similar questions as mine. However, most of them do not have a concrete explanation to what I'm looking for.
I understand that the Span of a Vector Space $V$ is the linear combination of all the vectors in $V$.
I also understand that the Basis of a Vector Space V is a set of vectors ${v_{1}, v_{2}, …, v_{n}}$ which is linearly independent and whose span is all of $V$.
Now, from my understanding the basis is a combination of vectors which are linearly independent, for example, $(1,0)$ and $(0,1)$.
But why?
The other question I have is, what do they mean by "whose span is all of $V$" ?
On a final note, I would really appreciate a good definition of Span and Basis along with a concrete example of each which will really help to reinforce my understanding.
Thanks.
Best Answer
Span is usually used for a set of vectors. The span of a set of vectors is the set of all linear combinations of these vectors.
So the span of $\{\begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}\}$ would be the set of all linear combinations of them, which is $\mathbb{R}^2$. The span of $\{\begin{pmatrix}2\\0\end{pmatrix}, \begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}\}$ is also $\mathbb{R}^2$, although we don't need $\begin{pmatrix}2\\0\end{pmatrix}$ to be so.
So both these two sets are said to be the spanning sets of $\mathbb{R}^2$.
However, only the first set $\{\begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}\}$ is a basis of $\mathbb{R}^2$, because the $\begin{pmatrix}2\\0\end{pmatrix}$ makes the second set linearly dependent.
Also, the set $\{\begin{pmatrix}2\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}\}$ can also be a basis for $\mathbb{R}^2$. Because its span is also $\mathbb{R}^2$ and it is linearly independent.
For another example, the span of the set $\{\begin{pmatrix}1\\1\end{pmatrix}\}$ is the set of all vectors in the form of $\begin{pmatrix}a\\a\end{pmatrix}$.