$ $M = \begin{array}{cc} x & y \\ z & w \\ \end{array}$ $Tricks for checking Linear Independence, Span and Basis
Linear Independence
If det(vectors) != 0 ⇔ linearly independent
If end result of the rref(vectors) gives you a matrix with all rows having leading 1's it is linearly independent.
If det(vectors) = 0 ⇔ linearly dependent
If end result of the rref(vectors) gives you a parameter in the equation, the vectors are linearly dependent.
Tip: If #No of vectors > Dimension ⇔ it is linearly dependent
Span
If Dimension > #No of vectors -> it CANNOT span
If det(vectors) != 0 ⇔ it spans
If end result of the rref(vectors) gives you a matrix with all rows having leading 1's, it spans.
If det(vectors) = 0 ⇔ does not span
If end result of the rref(vectors) gives you a matrix with not all rows having a leading 1, it does not span.
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
I'n's'e'r't'f'o'r'm'u'l'a'h'e'r'e
