WUSTL ESE 318 - Lecture 6 - Rank, Systems of Eqns (9 pages)

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Lecture 6 - Rank, Systems of Eqns



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Lecture 6 - Rank, Systems of Eqns

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Pages:
9
School:
Washington University in St. Louis
Course:
Ese 318 - Engineering Mathematics A

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ESE 318 02 Fall 2014 Lecture 6 Rank Systems of Eqns Determinants Sept 11 2014 Rank Formal definition The rank of an mXn matrix A denoted by rank A is the maximum number of linearly independent row vectors in A Computing rank You don t have to think through the definition to determine rank there is an easy way to compute rank Rank is the number of non zero rows in the rowechelon form To get a matrix into row echelon form use Gaussian elimination Note this applies to any matrix In fact every matrix has a rank and it can be computed using Gaussian elimination Assume we started with matrices and performed Gaussian elimination to transform them into row echelon equivalent forms as below The corresponding ranks are indicated as well as the number of variables unknowns which is n or columns of A 1 A B 0 0 1 A B 0 0 1 A B 0 0 0 A B 0 5 7 2 1 1 5 rank A rank A B 3 n 3 5 7 2 1 0 0 rank A rank A B 2 n 3 rank A 2 0 5 7 2 1 0 1 0 1 6 0 0 0 6 1 0 6 1 0 6 1 2 6 1 4 rank A B 3 n 3 rank A rank A B 2 n 5 The above matrices have augmented matrix notation but the concept of rank applies to any matrix Labor saving tip If all you are interested in is rank the matrix does not have to be in technical row echelon form the leading non zero entry in a row does not have to be 1 All we want is the number of non zero rows so any row could be multiplied by a nonzero scalar and the result would be the same So to find rank you might not have to do full Gaussian elimination Applying rank to characterize systems of equations We saw before that a system of equations can have no solution a unique solution or infinite solutions We can totally characterize the system of equations in this way simply from looking at rank A rank A B of the augmented matrix and the number of unknowns n where n also is the number of columns of A ESE 318 02 Fall 2014 The rules are easily seen in the form of a decision table Also Zill has a decision tree on page 464 I don t like Zill s tree so much though Here r A rank A



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