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WUSTL ESE 318 - Lecture 6 - Rank, Systems of Eqns

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ESE 318-02, Fall 2014Lecture 6: Rank, Systems of Eqns, DeterminantsSept. 11, 2014Rank. 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, thereis an “easy” way to compute rank: Rank is the number of non-zero rows in the row-echelon 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).            5,2|461000026100|3,3|,2117000210561|3,2|017000210561|3,3|517100210561|nBArankArankBAnBArankArankBAnBArankArankBAnBArankArankBA(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 non-zero 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 2014The 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).Decision Table for Determining Number of Solutions and Free Parametersr(A) = r(A|B) Y Y Nr(A) = n Y N either0 solutions -- -- X1 solution X -- --Infinite solutions -- n- r(A) --Examples. Look at the following row-echelon matrices as examples. Find rank(A), rank(A|B), and n, and determine: number of solutions and, if infinite solutions, number of arbitrary (free) parameters.            parametersarbitrarysolutionsinf.nBArankAranksolutionnontinconsisteBArankArankparameterarbitraryonesolutionsinf.nBArankAranksolutionuniquenBArankArank3,52|461000026100,3|,2117000210561,32|0170002105613|517100210561Homogeneous systems. See that for a homogeneous system, rank(A) = rank(A|B) always.We can use the above decision table to make a simpler one for homogeneous systems.Abbreviated Decision Table for Homogeneous Systemsrank(A) = n Y N1 solution Trivial solution --Infinite solutions -- n-rClarification. The terms “row-echelon”, “reduced row-echelon” and “rank” all apply to any matrix. For example, you can find the rank of any matrix by using Gaussian elimination and then counting the number of non-zero rows in the resulting row-echelon form.But, I occasionally take a short-cut when referring to augmented matrices when rank(A) <rank(A|B). Consider the following example.01170000002105617517000000210561| BAESE 318-02, Fall 2014The matrix on the left is not yet in row-echelon form, although the “A” part is. A little more Gaussian elimination results in the matrix on the right in true row-echelon form.Since (1) we usually are only concerned if rank(A) = rank(A|B) when dealing with augmented matrixes, and (2) we can do this last step in our heads, I often do not make it explicit. That might cause some confusion.Two facts about rank, and a tip.- By definition, the 0 matrix has rank = 0. All non-zero matrices have rank > 0.- The rank of an m X n matrix < minimum of m and n. That is, the rank is no biggerthan the smaller dimension of the matrix (rows or columns).- In using row reduction (Gaussian elimination) to find rank, you don’t have to make the leading non-zero terms 1. This could save you time.Examples:8.3.9. Find rank.3000000001103123134301121011031213430312110112103201266631215014224203432313431324232313431322133132313443231213132rankRRRRRRRRRRRESE 318-02, Fall 20148.2.8.    txtxxttxxxxxxtxtxxxthentxLetparameterfreesolutionsnBArArxxxxxx32132132122323321321421194422942942424221,32|4210942144221109421121594211259428.2.12.    tttxxxtxthentxLetparameterfreesolutionsnBArArxxxxxxxx212332123232331321321221,32|00000100211096009600211030605420211036054202ESE 318-02, Fall 20148.2.19.   solutionsntinconsisteBArArxxxxxxxxxxxxxxx0,|4310000141004111011531511103301041110115315111033010115314111062641245211153141110626424521534432143214321432The Determinant. (Note: this will be testable on Exam 2, not Exam 1.) Only a square matrix can have a determinant. A determinant is a scalar (a number) associated with a


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WUSTL ESE 318 - Lecture 6 - Rank, Systems of Eqns

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