Find the basis and dimension of the span of the given vectors: (1,0), (0,1), (1,1) and (1,2,0), (0,1,3).

Let the given vectors be $\vec{v_1} = (1,0)$, $\vec{v_2} = (0,1)$, and $\vec{v_3} = (1,1)$. These vectors are in $\mathbb{R}^2$.
We check for linear independence. The vectors $\vec{v_1}$ and $\vec{v_2}$ are the standard basis vectors for $\mathbb{R}^2$, so they are linearly independent.
We can express $\vec{v_3}$ as a linear combination of $\vec{v_1}$ and $\vec{v_2}$:

$$(1,1) = 1 \cdot (1,0) + 1 \cdot (0,1)$$

Since $\vec{v_3}$ is a linear combination of $\vec{v_1}$ and $\vec{v_2}$, it is linearly dependent on them.
Therefore, the set $\{(1,0), (0,1), (1,1)\}$ is linearly dependent.
A basis for the span is a maximal set of linearly independent vectors. In this case, $\{(1,0), (0,1)\}$ forms a basis.
The dimension of the span is the number of vectors in its basis.
The dimension of Span$\{(1,0), (0,1), (1,1)\}$ is 2.

The dimension of Span$\{(1,0), (0,1), (1,1)\}$ is $\boxed{\text{2}}$.

Let the given vectors be $\vec{u_1} = (1,2,0)$ and $\vec{u_2} = (0,1,3)$. These vectors are in $\mathbb{R}^3$.
To find the basis and dimension, we first check if the vectors are linearly independent.
Two vectors are linearly dependent if one is a scalar multiple of the other.
Assume $c \cdot \vec{u_1} = \vec{u_2}$ for some scalar $c$.

$$c(1,2,0) = (0,1,3)$$ $$(c, 2c, 0) = (0,1,3)$$

From the first component, $c=0$.
If $c=0$, then $(0,0,0) = (0,1,3)$, which is false.
Therefore, $\vec{u_1}$ and $\vec{u_2}$ are not scalar multiples of each other, meaning they are linearly independent.
Since the vectors are linearly independent and they span the given space, they form a basis for that space.
The basis for Span$\{(1,2,0), (0,1,3)\}$ is $\{(1,2,0), (0,1,3)\}$.
The dimension of the span is the number of vectors in its basis.
The dimension of Span$\{(1,2,0), (0,1,3)\}$ is 2.

The basis is $\boxed{\{(1,2,0), (0,1,3)\}}$ and the dimension is $\boxed{\text{2}}$.