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Unformatted text preview: MA 35100 LECTURE NOTES: FRIDAY, FEBRUARY 26 Redundant Vectors We use the example from the previous lecture to make a definition. Definition. Consider vectors vectorv 1 , vectorv 2 , . . . , vectorv m R n , and let V = span( vectorv 1 ,vectorv 2 , . . . ,vectorv m ) be a subspace of R n . a. A vector vectorv i is redundant if it is a linear combination of the other vectors i.e., the vector is redundant in terms of the span: V = span ( vectorv 1 , . . . ,vectorv i- 1 ,vectorv i +1 , . . . ,vectorv m ) . b. The vectors vectorv 1 ,vectorv 2 , . . . ,vectorv m are said to be linearly independent if none of them is redundant. Otherwise, we say the set is linearly dependent . c. The vectors vectorv 1 ,vectorv 2 , . . . ,vectorv m is said to form a basis for V if (i) they span V and (ii) they are linearly independent. In general, say that we have a linear transformation T : R m R n . The following algorithm computes a basis for im( T ): (1) Compute the matrix A of T : A = bracketleftbig vectorv 1 vectorv 2 vectorv m bracketrightbig where vectorv j = T ( vectore j ) R n . (2) Remove the redundant vectors v i from the list vectorv 1 ,vectorv 2 , . . . ,vectorv m . (3) The remaining vectors are a basis for im( T )....
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