Term
Equation Operations Preserve Solution Sets (EOPSS) |
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Definition
If we apply one of the three equation operations to a system of linear equations, then the original system and the transformed system are equivalent. |
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Term
Row-equivalent Matrices Represent Equivalent Systems (REMES) |
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Definition
Suppose A and B are row-equivalent augmented matrices. Then the systems of linear equations they represent are equivalent systems. |
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Term
Row Equivalent Matrix in Echelon Form (REMEF) |
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Definition
Suppose A is a matrix. Then there is a matrix B such that
- A and B are row-equivalent
- B is in row-reduced echelon form.
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Term
Reduced Row Echelon Form is Unique (RREFU) |
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Definition
Suppose that A is an m x n matrix and that B and C are m x n matrices that are row-equaivalent to A and in row reduced echelon form. Then B = C. |
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Term
Recognizing Consistency of a Linear System (RCLS) |
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Definition
Suppose A is the augmented matrix of a system of linear equations with n variables. Suppose also that B is a row-equivalent matrix in row-reduced echelon form with r non-zero rows.
The system of equations is inconsistent if and only if column (n+1) of B is a pivot column.
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Term
Consistent Systems, r and n (CSRN) |
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Definition
Suppose A is the augmented matrix of a system of linear equations with n variables. Suppose also that B is a row-equivalent matrix in row-reduced echelon form with r pivots. Then [image]. If [image], then the system has a unique solution and if [image], then the system has infinitely many solutions. |
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Term
Free Variables for Consistent Systems (FVCS) |
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Definition
Suppose A is the augmented matrix of a system of linear equations with n variables. Suppose also that B is a row-equivalent matrix in row-reduced echelon form with r row that are not completely zeros. Then the solution set can be described with n-r free variables. |
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Term
Possible Solution Sets for Linear Systems (PSSLS) |
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Definition
A system of linear equations has no solution, a unique solution or infinitely many solutions. |
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Term
Consistent, More Variables than Equations, Infinite (CMVEI) |
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Definition
Suppose a consistent system of linear equations has m equations in n variables. If n>m, then the system has infinitely many solutions. |
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Term
Homogeneous Systems Are Consistent (HSC) |
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Definition
Suppose that a system of linear equations is homogeneous. Then the system is consistent and one solution is found by setting each variable to zero. |
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Term
Homogeneous, More Variables than Equations, Inifinite Solutions (HMVEI) |
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Definition
Suppose that a homogeneous system of linear equations has m equations and n variables with n > m. Then the system has infinitely many solutions. |
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Term
Nonsingular Matrices Row Reduce to the Identity Matrix (NMRRI) |
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Definition
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. |
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Term
Nonsingular Matrices Have Trivial Null Spaces (NMTNS) |
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Definition
Suppose that A is a square matrix. Then A is nonsingular if and only if the null space of A is the set containing only the zero vector. i.e. N(A) = {0} |
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Term
Nonsingular Matrices And Unique Solutions (NMUS) |
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Definition
Suppose that A is a square matrix. A is a nonsingular matrix if and only if the system LS(A,b) has a unique solution for every choice of the constant vector b. |
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Term
Nonsingular Matrix Equivalences (NME) |
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Definition
Suppose that A is a square matrix. The following are equivalent:
- A is nonsingular.
- A row-reduces to the identity matrix.
- The null space of A contains only the zero vector, N(A)={0}.
- The linear system LS(A,b) has a unique solution for every possible choice of b.
- The columns of A form a linearly independent set.
- A is invertible.
- The column space of A is [image], [image]
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Term
Vector Space Properties of Column Vectors (VSPCV) |
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Definition
Suppose that [image] is the set of column vectors of size m with addition and scalar multiplication as defined in CVA and CVSM.
Then, if [image] and [image]
- ACC Additive Closure
[image]
- SCC Scalar Closure
[image]
- CC Commutivity
[image]
- AAC Additive Associativity
[image]
- ZC Zero Vector
[image]
- AIC Additive Inverse
[image]
- SMAC Scalar Multiplication Associativity
[image]
- DVAC Distributivity Across Vector Addition
[image]
- DSAC Distributivity across Scalar Multiplication
[image]
- OC One Column
[image]
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Term
Solutions to Linear Systems are Linear Combinations (SLSLC) |
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Definition
Denote the columns of the m x n matrix A as the vectors [image]. Then [image] is a solution to the linear system of equations LS(A,b) if and only if b equals the linear combination of the columns of A formed with the entities of x.
[image]
[Solutions to systems of equations are linear combinations of the n column vectors of the coefficient matrix (Aj) which yield the constant (column) vector b.] |
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Term
Vector Form of Solutions to Linear Systems (VFSLS) |
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Definition
Suppose that [A|b] is the augmented matrix for a consistent linear system LS(A,b) of m equations in n variables.
Let B be a row equivalent m x (n+1) matrix in reduced row-echelon form. Suppose that B has r pivot columns, with indices [image] while the n-r non-pivot columns have indices in [image].
Define vectors [image] of size n by
[image]
[image]
Then the set of solutions to the system of equations LS(A,b) is
[image]
[Note: the value of [image] in the solution set are the values of the free variables [image] ] |
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Term
Particular Solution Plus Homogeneous Solutions (PSPHS) |
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Definition
Suppose that [image] is one solution to the linear system of equations LS(A,b). Then [image] is a solution to LS(A,b) if and only if [image] for some vector [image]. |
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Term
Spanning Sets for Null Spaces (SSNS) |
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Definition
Suppose that A is an m x n matrix, and B is a row equivalent matrix in reduced row-echelon form.
Suppose that B has r pivot columns, with indices given by [image], while the n-r non-pivot column have indices [image]. Construct n-r vectors [image] of size n.
[image]
Then the null space of A is given by
[image] |
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Term
Linear Independence Vectors and Homogeneous Systems (LIVHS) |
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Definition
Suppose that [image] is a set of vectors and A is the m x n matrix whose columns are the vectors in S. Then S is a linearly independent set if and only if the homogeneous system LS(A,0) has a unique solution. |
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Term
Linearly Independent Vectors r and n (LIVRN) |
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Definition
Suppose that [image] is a set of vectors and A is an m x n matrix whose columns are the vectors in S. Let B be a matrix in reduced row-echelon form that is row equivalent to A and let r denote the number of pivot rows in B. Then S is linearly independent if and only if n == r. |
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Term
More Vectors than Size Implies Linear Independence (MVSLD) |
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Definition
Suppose that [image] and n > m. Then S is a linearly independent set. |
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Term
Nonsingular Matrices have Linearly Independent Columns (NMLIC) |
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Definition
Suppose that A is a square matrix. Then A is nonsingular if and only if the columns of A form a linearly independent set. |
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Term
Basis for Null Spaces (BNS) |
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Definition
Suppose that A is an m x n matrix and B is a row equivalent matrix in reduced row-echelon form with r pivot columns.
Let [image] and [image] be the set of column indices where B does and does not have pivot columns. Construct n-r vectors [image] of size n as
[image]
Define the set [image]. Then
- N(A) = (S)
- S is a linearly independent set.
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Term
Dependency in Linearly Dependent Sets (DLDS) |
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Definition
Suppose that [image] is a set of vectors. Then S is a linearly dependent set if and only if there is an index t, [image] such that [image] is a linear combination of the vectors [image]. |
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Term
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Definition
Suppose that [image] is a set of column vectors. Define [image] and let A be the matrix whose columns are vectors from S. Let B be the reduced row-echelon form of A, with [image] the set of indices for the pivot columns of B. Then
- [image] is a linearly independent set
[image]
(The minimum span of a set is made up of the column vectors that correspond to the pivot columns which in turn correspond to the dependent variables.) |
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Term
Conjugation Respects Vector Addition (CRVA) |
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Definition
Suppose x and y are two vectors from [image]. Then
[image]
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Term
Conjugation Respects Vector Scalar Multiplication (CRSM) |
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Definition
Suppose x is a vector from [image], and [image] is a scalar. Then
[image]
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Term
Inner Product and Vector Addition (IPVA) |
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Definition
Suppose [image]. Then
- [image]
- [image]
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Term
Inner Product and Scalar Multiplication (IPSM) |
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Definition
Suppose [image]. Then
- [image]
- [image]
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Term
Inner Product as Anti-Commutative (IPAC) |
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Definition
Suppose that u and v are vectors in [image]. Then
[image] |
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Term
Inner Products and Norms (IPN) |
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Definition
Suppose that u is a vector in [image]. Then
[image] |
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Term
Positive Inner Products (PIP) |
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Definition
Suppose that u is a vector in [image]. Then
[image] |
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Term
Orthogonal Sets are Linearly Independent (OSLI) |
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Definition
Suppose that S is an orthogonal set of nonzero vectors. Then S is linearly independent. |
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Term
Gram-Schmidt Procedure (GSP) |
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Definition
Suppose that [image] is a linearly independent set of vectors in [image]. Define the vectors [image] by
[image]
Let [image]. Then T is an orthogonal set of nonzero vectors and [image]. |
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