C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes1-7

C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes1-7 - v...

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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 1. A list consisting only of 0 is LD. 2. A list of two vectors is LD if one is a multiple of the other. 3. A list of more than two vectors if LD if either it contains 0 or if one of the vectors is a LC of the others in the list. LINEAR DEPENDENCE RELATIONS A list of vectors is LD if and only if there is a LC with weights not all zero for which LC = 0 . This equation is called a linear dependence relation.
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A list of vectors is linearly independent (LI) if the list is NOT linearly dependent. A list of vectors
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Unformatted text preview: v 1 , v 2 , . . . , v p is LI if and only if the vector equation x 1 v 1 + x 2 v 2 + + x p v p = has ONLY the trivial solution. EXAMPLE. Are the vectors 1-3 2 , 2-5 3 , and 4 1 linearly independent? Justify your answer. Any list of vectors containing the zero vector is LD. Why? Any list of vectors with more vectors than rows is LD. Why? If one vector in a list is a multiple of another vector in the list, then the list is LD. Why? How many pivot columns must a 9 6 matrix have if its columns are linearly independent? Why? HOMEWORK: SECTION 1.7...
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This note was uploaded on 04/15/2008 for the course M 340L taught by Professor Pavlovic during the Spring '08 term at University of Texas at Austin.

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