Theorems for Matrix Theory

Matrix theory theorems

19 cards | Total Attempts: 182 | Created by shugo000 | Last updated: Sep 18, 2015 | Total Attempts: 182

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Theorem EOPSS - Equation Operations Preserve Solution Sets

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Cards In This Set

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Theorem EOPSS - Equation Operations Preserve Solution Sets	If we apply one of the three equation operations of Definition EO to a linear system, then the original system and the transformed system are equivalent.
Theorem REMES - Row-Equivalent Matrices represent Equivalent Systems	Suppose that A and B are row-equivalent augmented matrices. Then the systems of linear equations that they represent are equivalent.
Theorem REMEF - Row-Equivalent Matrix in Echelon Form	Let A be any matrix. Then there exists a matrix B such that 1) A and B are row-equivalent. 2) B is in reduced row-echelon form.
Theorem RREFU - Reduced Row-Echelon Form is Unique	Suppose that A is an m X n matrix and that B and C are m X n matrices that are row-equivalent to A and in reduced row-echelon form. Then B = C.
Theorem RCLS - Recognizing Consistency of a Linear System	Suppose A is the augmented matrix of a linear system with n variables. Suppose also that B is a row-equivalent matrix in reduced row-echelon form with r nonzero rows. Then the system is inconsistent if and only if the leading 1 of row r is located in column n + 1 of B.
Theorem RCLS - Recognizing Consistency of a Linear System	Suppose A is the augmented matrix of a linear system in n variables. Suppose also that B is a row-equivalent matrix in reduced row-echelon form with r nonzero rows. If r = n + 1, then the system is inconsistent.
Theorem CSRN - Consistent Systems, r and n	Suppose that A is the augmented matrix of a consistent linear system with n variables. Suppose also that B is a row-equivalent matrix in reduced row-echelon form with r nonzero rows. Then r 6 n. Moreover, if r = n, then the system has a unique solution; if r < n, then the system has infinitely many solutions.
Theorem FVCS - Free Variables for Consistent Systems	Suppose that A is the augmented matrix of a consistent linear system with n variables. Suppose also that B is a row-equivalent matrix in reduced row-echelon form with r nonzero rows. Then the solution set can be described with n - r free variables.
Theorem PSSLS - Possible Solution Sets for Linear Systems	A system of linear equations has no solutions, a unique solution, or infinitely many solutions.
Theorem CMVEI - Consistent, More Variables than Equations, Infinite solutions	A consistent linear system with more variables than equations has infinitely many solutions.
Theorem HSC - Homogeneous Systems are Consistent	A homogeneous linear system is consistent.
Theorem HMVEI - Homogeneous, More Variables than Equations, infinite solutions	Suppose that a homogeneous system of linear equations has m equations and n variables with n > m. Then the system has infinitely many solutions.
Theorem NMRRI - Nonsingular Matrices Row Reduce to the Identity matrix	Suppose that A is a square matrix and B is a row-equivalent matrix in reduced row-echelon form. Then A is nonsingular if and only if B is the identity matrix.
Theorem NMTNS - Nonsingular Matrices have Trivial Null Spaces	A square matrix A is nonsingular if and only if N(A) = {0}.
Theorem NMUS - Nonsingular Matrices and Unique Solutions	A square matrix A is nonsingular if and only if the system LS(A, b) has a unique solution for every choice of the constant vector b.