Understanding the difference between Span and Basis

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.

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 $\begin{pmatrix}a\\a\end{pmatrix}$.

The basis is a combination of vectors which are linearly independent and which spans the whole vector V.

Suppose we take a system of $R^2$. Now as you said,$(1,0)$ and $(0,1)$ are the basis in this system and we want to find any $(x,y)$ in this system.

$$ (x,y)=x(1,0)+y(0,1) $$ where $x$ , $y$ belong to a set of real numbers.

Using the linear combination of of the basis , you can find find any vector in the system.

For instance, $$ (98745,12345)=98745(1,0)+12345(0,1) $$

Now , what is span?

Span is the set of all linear combination vectors in the system.

In $R^2$,suppose span is the set of all combinations of $(1,0)$ and $(0,1)$.

This set would contain all the vectors lying in $R^2$,so we say it contains all of vector V.

Therefore, 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.

The span of a finite subset $S$ of a vector space $V$ is the smallest subvector space that contains all vectors in $S$. One shows easily it is the set of all linear combinations of lelements of $S$ with coefficients in the base field (usually $\mathbf R,\mathbf C$ or a finite field).

A basis of the vector space $V$ is a subset of linearly independent vectors that span the whole of $V$. If $S=\{x_1, \dots, x_n\}$ this means that for any vector $u\in V$, there exists a unique system of coefficients such that $$u=\lambda_1 x_1+\dots+\lambda_n x_n. $$