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| − | '''Tricks for checking Linear Independence, Span and Basis''' | + | '''<math>$M = |
| + | \begin{array}{cc} | ||
| + | x & y \\ | ||
| + | z & w \\ | ||
| + | \end{array}$</math>Tricks for checking Linear Independence, Span and Basis''' | ||
| + | <br> | ||
| + | <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'''. | + | |
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.''' | ||
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If Dimension > #No of vectors -> '''it CANNOT span''' | If Dimension > #No of vectors -> '''it CANNOT span''' | ||
| − | 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'''. | + | 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'''. |
| − | 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.''' | + | 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> | <u>'''Basis'''</u><br> | ||
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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> <span class="texhtml">''I''''n''''s''''e''''r''''t''''f''''o''''r''''m''''u''''l''''a''''h''''e''''r''''e''</span> |
[[Category:MA265Spring2011Momin]] | [[Category:MA265Spring2011Momin]] | ||
Revision as of 08:20, 1 May 2011
$ $M = \begin{array}{cc} x & y \\ z & w \\ \end{array}$ $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
