| Line 1: | Line 1: | ||
| − | ''' | + | '''Tricks for checking Linear Independence, Span and Basis''' |
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
<br> <u>'''Linear Independence'''</u> | <br> <u>'''Linear Independence'''</u> | ||
| − | If det(vectors) != 0 ⇔ '''linearly independent'''<br> 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 independent'''<br> If end result of the rref(vectors) gives you a matrix with all rows having leading 1's it is '''linearly independent'''. <math>$M = \left( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \right)$</math> |
If det(vectors) = 0 ⇔ '''linearly dependent'''<br>If end result of the rref(vectors) gives you a parameter in the equation, the vectors are '''linearly dependent.''' | If det(vectors) = 0 ⇔ '''linearly dependent'''<br>If end result of the rref(vectors) gives you a parameter in the equation, the vectors are '''linearly dependent.''' | ||
Revision as of 07:22, 1 May 2011
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. $ $M = \left( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \right)$ $
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
