Is the concept of vector size something relative to the vector space? The vector $\begin{pmatrix}1 \\ 0\end{pmatrix}$ will always have this size: $\sqrt{1^2+0^2}=\sqrt{1}=1$. But given the basis $\begin{bmatrix}1 & 1 \\ 1 & 0\end{bmatrix}$, the coordinate vector $\begin{pmatrix}1 \\ 0\end{pmatrix}$ is pointing to the point $\begin{pmatrix}1 \\ 1\end{pmatrix}$ in the standard basis, so its size can also be calculated this way: $\sqrt{1^2+1^2}=\sqrt{2}$ , so maybe I'm making a silly mistake here but I can't find it out. Could a size such as $1$ in a vector space have a different value (e.g. $\sqrt{2}$) in another vector space? Is it illegal to interpret things in the standard space when we're in a different vector space?
 A: A vector doesn't a priori have a size. Abstractly, a vector $v$ is just an element of a vector space $V$, which is a set with some operations (addition, multiplication by a scalar) and rules about how those operations work (e.g. commutativity).
If you want to talk about vectors with sizes, you are talking about a normed vector space. A normed vector space is just a vector space $V$ with a function $\| \cdot \|_V$ defined on it, called the norm. For any vector $v \in V$, you get a real number $\| v \|_V$, and it is supposed to satisfy the following three rules:

*

*For any $v \in V$, $\|v\|_V \geq 0$.

*If $\|v\|_V = 0$, then $v = 0$.

*If $\alpha$ is any scalar, then $\|\alpha \cdot v\|_V = |\alpha|\cdot \|v\|_V$

*For any $v, w \in V$, $\|v + w\|_V \leq \|v\|_V + \|w\|_V$.

You already know about at least one norm: the standard Euclidean norm on the vector space $\mathbb R^2$, where
$$
\left\|\begin{pmatrix}a\\b\end{pmatrix}\right\|_2 = \sqrt{a^2 + b^2}.
$$
(This 2 in the bottom specifies that this is the Euclidean norm you are talking about.)
You can check that this satisfies all the rules listed above. But you could define lots of other norms on the vector space $\mathbb R^2$; for example, $\|v\|_\infty = \max\{|v_1|, |v_2|\}$, or $\|v\|_{\text{skew}} = \sqrt{(v_1+v_2)^2 + v_1^2}$. In fact that last one is exactly the norm you have calculated by looking at your vectors in a different basis.
In general, if you have a change of basis, this will usually not preserve the norm. (In fact, changes of basis that preserve the norm are quite special.) That is, by changing the basis you get a new norm, and when specifying what you mean by "size" of your vector, you need to be clear about what norm you are using if it isn't clear from context.
