Linear Independence (9/26/05) A set of vectors {v 1, v 2, …, v n } is said to be linearly independent if the homogeneous vector equation x 1 v 1 + x 2.

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Presentation transcript:

Linear Independence (9/26/05) A set of vectors {v 1, v 2, …, v n } is said to be linearly independent if the homogeneous vector equation x 1 v 1 + x 2 v 2 + … + x n v n = 0 has only the trivial solution. Otherwise the set is said to be linearly dependent, meaning that at least one of the vectors can be written as a linear combination of the others.

Examples of Independence and Dependence A single non-zero vector is independent. Two non-zero vectors are linearly independent if (what??). Three non-zero vectors are linearly independent if none of them lies in the span of the other two. Use this last statement to formulate a statement about n vectors being linearly independent.

A Criterion for Dependence The set of non-zero vectors {v 1, v 2, …, v n } is linearly dependent if for some index j > 1, v j can be written as a linear combination of the vectors {v 1, v 2, …, v j -1 }. Said another way, v j lies in the span of {v 1, v 2, …, v j -1 }. Note that this is certain to happen if the number of vectors exceeds the dimension of the vectors. (Why? Think about span.)

Spanning Versus Independence Generally speaking, spanning and linear independence run in opposite directions. That is, for a fixed vector dimension (say n) The fewer vectors there are, the more likely the set is independent. The more vectors there are, the more likely it is R n is spanned by the set If there are exactly n vectors, we’ll having spanning of R n iff we have independence.

Assignment for Wednesday Hand-in #1 is due at classtime. Read Section 1.7. In that section do the Practice and Exercises 1, 3, 5, 7, 9, 15, 19, 21, 23, 24, 27, 28.