Linear Dependence/Independence

[Gear300]

It is stated that for n-dimensional Euclidean space, n vectors are needed at least for linear independence. But if an n-dimensional Euclidean space also includes (n-1)-dimensional Euclidean space, then why can't it also include a family of n-1 linearly independent vectors?

 

[Petek]

It is stated that for n-dimensional Euclidean space, n vectors are needed at least for linear independence. But if an n-dimensional Euclidean space also includes (n-1)-dimensional Euclidean space, then why can't it also include a family of n-1 linearly independent vectors?

It can. For example, in three dimensional Euclidean space the single vector (1,0,0) is linearly independent, as are the two vectors {(1,0,0), (0,1,0)} and the three vectors {(1,0,0), (0,1,0), (0,0,1)}. However, any four vectors in this space will necessarily be linearly dependent. Does that help?

 

[Gear300]

I get what you're saying. I mixed stuff up (thought too hard about the words "at least"). Thanks.

 

[HallsofIvy]

More than n vectors cannot be independent in Rⁿ. Less can be.