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'''Tricks for checking Linear Independence, Span and Basis''' | '''Tricks for checking Linear Independence, Span and Basis''' | ||
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| + | Note: This article assumes that you can only calculate a det for a square matrix. | ||
<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 | + | If det(vectors) != 0 ⇔ '''linearly independent'''<br> If end result of the rref(vectors) gives an identity matrix, 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'''. For example: <math>rref(\left( \begin{smallmatrix} 1&2&3\\ 2&3&4 \end{smallmatrix} \right)$) = \left( \begin{smallmatrix} 1&0&-1\\ 0&1&2 \end{smallmatrix} \right)$</math><br> |
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.''' | ||
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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 | + | If #No of vectors = Dimension -> it has to be linearly independent to span<span class="texhtml">'' |
| + | ''</span> | ||
[[Category:MA265Spring2011Momin]] | [[Category:MA265Spring2011Momin]] | ||
Revision as of 07:30, 1 May 2011
Tricks for checking Linear Independence, Span and Basis
Note: This article assumes that you can only calculate a det for a square matrix.
Linear Independence
If det(vectors) != 0 ⇔ linearly independent
If end result of the rref(vectors) gives an identity matrix, 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. For example: $ rref(\left( \begin{smallmatrix} 1&2&3\\ 2&3&4 \end{smallmatrix} \right)$) = \left( \begin{smallmatrix} 1&0&-1\\ 0&1&2 \end{smallmatrix} \right)$ $
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
