What does it mean to say that a vector is 'coordinate invariant'?

In my lecture notes, it says that a vector is 'coordinate invariant' if it's properties do not depend on the choice of basis used to represent them.

I understand that the basis of a vector space is a set of linearly independent vectors which span the vector space, but I'm struggling to interpret the given definition of coordinate invariance. If anyone could clear this up for me, perhaps with an example, that would be greatly appreciated.

• What class is this for? Is this a linear algebra class? – Omnomnomnom Dec 28 '15 at 14:14
• Vector calculus. – M Smith Dec 28 '15 at 14:19
• Is it possible your lecturer meant something like "a property of a vector is 'coordinate invariant' if the property does not depend on the choice of basis..."? If that's the case, then geometric properties of a vector $v$, such as Cartesian magnitude of $v$, or the angles between $v$ and the Cartesian axes, are coordinate invariant, while the components of $v$ are not coordinate invariant. – Andrew D. Hwang Dec 28 '15 at 14:40

For example, $v+w$ is a coordinate-invariant expression, because vector addition is the same no matter which coordinate system we work in. Conversely, $(v_1+w_1, v_2+w_2, v_3+w_3)$ is not coordinate-invariant, because picking out particular coordinates of the vectors depends on which coordinate system you're using.