A set of two basis vectors couldn't possibly span R4. Are you sure that what you saw wasn't talking about a subspace of R4?grassstrip1 said:Hey everyone I just had a quick thought that was bothering me. For a set to be a basis it must be linearly independent and span the vector space. I've seen cases however of only two vectors forming a basis for R4 I don't see how two vectors could span 4 space or am I missing something.
Thanks!
A linearly independent set in R4 is a set of vectors in four-dimensional space that cannot be written as a linear combination of other vectors in the same set.
To determine if a set of vectors is linearly independent in R4, you can create an augmented matrix with the vectors as columns and use row reduction to see if the matrix has a unique solution. If it does, the vectors are linearly independent. If not, they are linearly dependent.
The span of a set of vectors in R4 is the set of all possible linear combinations of those vectors. In other words, it is the set of all vectors that can be created by scaling and adding the original vectors in the set.
If a set of vectors is linearly independent in R4, then its span is equal to the entire vector space of R4. This means that any vector in R4 can be written as a linear combination of the vectors in the linearly independent set.
Yes, it is possible for a set of vectors in R4 to be linearly independent and span only a subspace of R4. This would occur if the set of vectors is not large enough to span the entire vector space, but the vectors within the set are not dependent on each other.