Basis of 4-dimensional space?

Question

Mathematics

I think and I can't figure it out. If we have a basis of a 4-dimensional space, can we say that if we have a 3-dimensional subspace we can express any 3-dimensional vector through the basis of our 4-dimensional space? (like if 4 is the coordinate 0 , is it still a vector of 3-dimensional space?)

Answer 1

Oh sure. This is always true.
But for rigor it is necessary to clarify. You write:

if 4 is the coordinate 0 , it's still a vector of 3-dimensional space

This is also true, but only if you choose the 3D subspace to be flat and perpendicular to the fourth basis vector.

Answer 2

I think we need to make an analogy with a 2D plane and 3D space. Any vector in a plane can be expressed in terms of space coordinates, but information is also needed on how the plane is located in space (4 numbers, for example, the normal to the plane and the distance to the center of coordinates).