is a linear combination of depends upon and
is a linear combination of depends upon and is called a DEPENDENT set.
When one vector in a set is a linear combination of other vectors in the set, then the set is said to be DEPENDENT. Suppose Let’s say is dependent. One of the vectors, , is a linear combination of the others: A nontrivial linear combination of the vectors produces the zero vector. At least one coefficient is not 0.
When one vector in a set is a linear combination of other vectors in the set, then the set is said to be DEPENDENT. Suppose Let’s say is dependent. One of the vectors, , is a linear combination of the others: A nontrivial linear combination of the vectors produces the zero vector. At least one coefficient is not 0.
Is the set dependent?
Is the set dependent? This is TRIVIAL. All coefficients are 0’s.
Is the set dependent? Is there any way to do this without using ALL ZEROS? NO NO is an INDEPENDENT SET
definition: is an INDEPENDENT set iff ONLY IF The ONLY linear combination of the vectors to produce the zero vector is the TRIVIAL one.
Suppose one of these is not 0. Let’s say c 2 0 DEPENDS on the other vectors in the set!
Is the set independent?
independent? Is the set Reduces to
independent? Is the set c 1 = 1 c 1 = c 3 c 2 = -3 c 3 Let c 3 = 1
independent? Is the set 1 -3 1 c 1 = c 3 c 2 = -3 c 3 Let c 3 = 1 NO
independent? Is the set 1 -3 1 1 3 -1 NO
Is the set independent?
independent? Is the set Reduces to
independent? Is the set ONLY IF c 1 = 0 c 2 = 0 c 3 = 0 YES
- Slides: 17