| Line 7: | Line 7: | ||
*If end result of the rref(vectors) gives an identity matrix, it is '''linearly independent'''<br> | *If end result of the rref(vectors) gives an identity matrix, it is '''linearly independent'''<br> | ||
*If end result of the rref(vectors) gives you a parameter in the matrix, the vectors are '''linearly dependent.''' | *If end result of the rref(vectors) gives you a parameter in the matrix, the vectors are '''linearly dependent.''' | ||
| − | |||
| − | |||
Tricks: | Tricks: | ||
| Line 24: | Line 22: | ||
If Dimension > #No of vectors -> '''it CANNOT span''' | If Dimension > #No of vectors -> '''it CANNOT span''' | ||
| − | + | *If end result of the rref(vectors) gives you a matrix with all rows having leading 1's, '''it spans'''.  | |
| + | *If end result of the rref(vectors) gives you a matrix with not all rows having a leading 1, it '''does not span.''' | ||
| + | |||
| + | Tricks: | ||
| + | If det(vectors) != 0 ⇔ it spans<br> | ||
| + | If det(vectors) = 0 ⇔ '''does not span'''<br> | ||
| − | + | 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> spans R<sup>2</sup><br> | |
<u>'''Basis'''</u><br> | <u>'''Basis'''</u><br> | ||
| Line 34: | Line 37: | ||
If #No of vectors > Dimension -> it is not a basis. | If #No of vectors > Dimension -> it is not a basis. | ||
| − | 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 |
[[Category:MA265Spring2011Momin]] | [[Category:MA265Spring2011Momin]] | ||
<br> | <br> | ||
Revision as of 09:41, 1 May 2011
Tricks for checking Linear Independence, Span and Basis
Note: For this article, I am assuming number of vectors is equal to the dimension of the vector space for calculating the determinant. If it is not, you need to do rref.
Linear Independence
- If end result of the rref(vectors) gives an identity matrix, it is linearly independent
- If end result of the rref(vectors) gives you a parameter in the matrix, the vectors are linearly dependent.
Tricks:
If det(vectors) != 0 ⇔ linearly independent
If det(vectors) = 0 ⇔ linearly dependent
If #No of vectors > Dimension ⇔ it is linearly dependent
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) $ is linearly dependent in R2 because the last column [-1 2]T i.e z is a parameter as there can be no leading 1 for that column. You can express x = z and y = -2z
Span
If Dimension > #No of vectors -> it CANNOT span
- If end result of the rref(vectors) gives you a matrix with all rows having leading 1's, it spans. 
- If end result of the rref(vectors) gives you a matrix with not all rows having a leading 1, it does not span.
Tricks:
If det(vectors) != 0 ⇔ it spans
If det(vectors) = 0 ⇔ does not span
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) $ spans R2
Basis
If Dimension > #No of vectors ⇔ cannot span ⇔ is not a basis
If #No of vectors > Dimension -> it is not a basis.
If #No of vectors = Dimension -> it has to be linearly independent to span
