is a linear combination of and depends upon and is called a DEPENDENT set.

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Presentation transcript:

is a linear combination of and 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 is dependent. One of the vectors, Let’s say, 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 is dependent. One of the vectors, Let’s say, 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 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?

Is the set independent? Reduces to

Is the set independent? c 1 = c 3 c 2 = -3c 3 Let c 3 = 1 c 2 = -3 c 1 = 1

Is the set independent? c 1 = c 3 c 2 = -3c 3 Let c 3 = 1 c 2 = -3 c 1 = NO

Is the set independent? 11-3 NO 13

Is the set independent?

Is the set independent? Reduces to

Is the set independent? c 1 = 0 c 2 = 0 c 3 = 0 ONLY IF YES