Difference between revisions of "????" - Rhea

Revision as of 07:15, 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

@@ Line 1: / Line 1: @@
 '''Tricks for checking Linear Independence, Span and Basis'''
-<u>'''Linear Independence'''</u>
-If det(vectors)&nbsp;!= 0 ⇔ '''linearly independent'''<br>If end result of the rref(vectors) gives [0 0 0 1] -&gt; it is linearly independent because the system is '''inconsistent'''
-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.'''
-Tip: If #No of vectors &gt; Dimension ⇔ it is '''linearly dependent'''<br>
+<u>'''Linear Independence'''</u>
-<u>'''Span'''</u><br>Tip: If the question asks if a vector “belongs to span” of other vectors, then it means it is asking if it’s '''linearly dependent'''
+If det(vectors)&nbsp;!= 0 ⇔ '''linearly independent'''<br>
-If Dimension &gt; #No of vectors -&gt; '''it CANNOT span'''
+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 det(vectors)&nbsp;!= 0 ⇔ it spans<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 '''span.'''
+Tip: If #No of vectors &gt; Dimension ⇔ it is '''linearly dependent'''<br>
-If det(vectors) = 0 ⇔ '''does not span'''<br>If end result of the rref(vectors) gives [0 0 0 | 1] -&gt; it does not span because the system is inconsistent
+<u>'''Span'''</u>
-<u>'''Basis'''</u><br>
+If Dimension &gt; #No of vectors -&gt; '''it CANNOT span'''
+If det(vectors)&nbsp;!= 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.'''
+<u>'''Basis'''</u><br>
 <br>If Dimension &gt; #No of vectors ⇔ cannot span ⇔ is not a basis
@@ Line 23: / Line 25: @@
 If #No of vectors &gt; Dimension -&gt; it has to be linearly dependent to span (check the tip)
 If #No of vectors = Dimension -&gt; it has to be linearly independent to span<br>
 [[Category:MA265Spring2011Momin]]

Difference between revisions of "????" - Rhea

Revision as of 07:15, 1 May 2011

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