Question

what is the reason behind it.

in vector spaces if v is the vector space and v={o} then the dimension of v = 0?

Answer 1

What's the definition of the dimension of a vector space? It is the number of basis vectors for that space.

How many basis vectors does V = {0} have?

Answer 2

i dont know

Answer 3

Zero.

Answer 4

Zero vector direction is arbitary

Answer 5

Zero vector directionand any vector parallel ,vertical

Answer 6

did this help?

Answer 7

@jamesj what do u mean by no. of basis vectors for v={0} is zero? means no vector in the basis of v={0}???

Answer 8

By convention and definition, a basis vector is always a non-zero vector. Hence the vector space V = {0}, a vector space that has only one vector, the zero vector, cannot has any basis vectors at all.

This is also accords with our geometric intuition about dimension. Three dimensional geometric space, the Cartesian space that you usually think about when you move about the world, has three dimensions: left-right, forward-backward, up-down. In vector space formalism, these three dimensions correspond to three basis vectors, one each in the x, y and z directions.

In the plane, there are two directions, two basis vectors and hence it is two dimensional.

A line only has one way you can move: backward-forward or left-right. So one basis vector, one dimension.

A point has no ways you can move. Hence zero dimensions.

Answer 9

it means v={6} has one dimension right?

Answer 10

V = {6} is not a vector space, because -6 is not a member of V.