notes1-7 - SECTION 1.7 LINEAR INDEPENDENCE We know that 3...

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SECTION 1.7 LINEAR INDEPENDENCE We know that 3 vectors in R 4 cannot span R 4 . What about 4 vectors, say v 1 , v 2 , v 3 , v 4 ? If, say, v 4 is a LC of the others, then what can we say about Span { v 1 , v 2 , v 3 , v 4 } ? LINEARLY DEPENDENT LISTS OF VECTORS. A list of vectors is linearly de- pendent if we can throw one of the vectors away, and the span of the resulting smaller list is the same as the span of the original list. In addition we say that a list consisting only of the zero vector is linearly dependent. WHAT ABOUT A LIST OF TWO VECTORS? WHAT ABOUT A LIST OF, SAY, p VECTORS? A list of two or more vectors is linearly dependent exactly when one of the vectors in the list is a linear combination of the others in the list.
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Rewrite the linear dependence property in terms of linear combinations of all the vectors in the list. The linear dependence of a list of two or more vectors can be expressed in three equivalent ways.
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