A basis in 2-space is a set of vectors which

• span all of 2-space
• are linearly independent.

You know already that you need at least two non-parallel vectors to span all of 2-space, and that any more than two vectors in 2-space cannot be linear independent, so any basis in 2-space must consist of two non-parallel vectors.

A basis in 3-space is a set of vectors which

• span all of 3-space
• are linearly independent.

You know already that you need at least three non-coplanar vectors to span all of 3-space, and that any more than three vectors in 3-space cannot be linear independent, so any basis in 3-space must consist of three non-coplanar vectors.

Notice that the number of vectors in a basis for a space is the "dimension" of that space: a basis in 2-space contains two vectors; a basis in 3-space contains three vectors. You could also talk about "1-space" - think of a line, which is spanned by a single non-zero vector.

The bases we've been using all along are referred to as standard bases, i.e.

The standard basis for 2-space is {i, j}.

The standard basis for 3-space is {i, j, k}.

Example. {i + 2j, i - j} is a basis of 2-space, since the vectors are not parallel, but {i - 2j, -2i + 4j} is not a basis of 2-space (the second vector is -2 times the first, so they are parallel).

Example. {i, i + j, i + j + k} is a basis of 3-space, since the three vectors don't all lie in some plane but the set of vectors
{i + 2j + 3k, 4i + 5j + 6k, 7i + 8j + 9k} is not a basis for 3-space, since all three vectors lie in the plane through the origin with normal vector [1, -2, 1].