MAT1341-L10-Appl-Vect-Spc

MAT1341-L10-Appl-Vect-Spc - APPLICATIONS OF GAUSSIAN...

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APPLICATIONS OF GAUSSIAN ELIMINATION TO VECTOR SPACES JOS ´ E MALAG ´ ON-L ´ OPEZ Recall that many basic questions about vector spaces can be stated in terms of linear combinations. It is not a surprise that we can obtain specific methods using Gaussian elimination. Basic Problems on Linear Combinations Let S = { v 1 , . . . , v r } be a fixed set of vectors in a vector space V , where V is either R n , P n , or M m × n . We can do the following: (1) Determine if a vector w is in Span { S } . If it is, we can obtain the coefficients that give us w as a linear combination of the vectors in S : (a) Consider the augmented matrix ( A | w ), where the i th col- umn of A is given by the coefficients/entries in v i . (b) Apply Gaussian elimination to ( A | w ). (c) If the obtained system is inconsistent, then w is not in Span { S } . (d) If the obtained system is consistent, then w can be written as w = α 1 v 1 + · · · + α r v r , where α 1 . . . α t is a solution of the system. Remark 1 . We can do this process simultaneously for more than one vector w . See the examples below. 1
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Example 2. Determine if the vectors 5 1 20 1 14 , 3 6 12 6 3 are in W = Span 1 2 4 2 1 , 4 9 16 9 3 , 5 8 20 9 6 Applying Gaussian elimination to the augmented matrix 1 4 5 5 3 2 9 8 1 6 4 16 20 20 12 2 9 9 1 6 1 3 6 14 3 we obtain 1 4 5 5
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