Tricks for checking Linear Independence, Span and Basis
Linear Independence
If det(vectors) != 0 ⇔ linearly independent
If end result of the rref(vectors) gives [0 0 0 1] -> it is linearly independent because the system is inconsistent
If det(vectors) = 0 ⇔ linearly dependent
If end result of the rref(vectors) is in the order of [0 0 0 0] and provided the system is consistent, the vectors are linearly dependent.
Tip: If #No of vectors > Dimension ⇔ it is linearly dependent
Span
Tip: If the question asks if a vector “belongs to span” of other vectors, then it means it is asking if it’s linearly dependent
If Dimension > #No of vectors -> it CANNOT span
If det(vectors) != 0 ⇔ it spans
If end result of the rref(vectors) is in the order of [0 0 0 | 0] and provided the system is consistent, the vectors span.
If det(vectors) = 0 ⇔ does not span
If end result of the rref(vectors) gives [0 0 0 | 1] -> it does not span because the system is inconsistent
Basis
If Dimension > #No of vectors ⇔ cannot span ⇔ is not a basis
If #No of vectors > Dimension -> it has to be linearly dependent to span (check the tip)
If #No of vectors = Dimension -> it has to be linearly independent to span
