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| − | '''Tricks for checking Linear Independence, Span and Basis''' | + | '''Tricks for checking Linear Independence, Span and Basis''' |
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| − | + | <u>'''Linear Independence'''</u> | |
| − | + | If det(vectors) != 0 ⇔ '''linearly independent'''<br> | |
| − | If | + | If det(vectors) = 0 ⇔ '''linearly dependent'''<br>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.''' |
| − | If | + | Tip: If #No of vectors > Dimension ⇔ it is '''linearly dependent'''<br> |
| − | + | <u>'''Span'''</u> | |
| − | <u>'''Basis'''</u><br> | + | If Dimension > #No of vectors -> '''it CANNOT span''' |
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| + | If det(vectors) != 0 ⇔ it spans<br>If end result of the rref(vectors) gives you a matrix with all rows having leading 1's, '''it spans'''. | ||
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| + | If det(vectors) = 0 ⇔ '''does not span'''<br>If end result of the rref(vectors) gives you a matrix with not all rows having a leading 1, it '''does not span.''' | ||
| + | <u>'''Basis'''</u><br> | ||
<br>If Dimension > #No of vectors ⇔ cannot span ⇔ is not a basis | <br>If Dimension > #No of vectors ⇔ cannot span ⇔ is not a basis | ||
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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 dependent to span (check the tip) | ||
| − | If #No of vectors = Dimension -> it has to be linearly independent to span<br> | + | If #No of vectors = Dimension -> it has to be linearly independent to span<br> |
[[Category:MA265Spring2011Momin]] | [[Category:MA265Spring2011Momin]] | ||
Revision as of 08:15, 1 May 2011
Tricks for checking Linear Independence, Span and Basis
Linear Independence
If det(vectors) != 0 ⇔ linearly independent
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
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
