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Linear Algebra Theorems
From 'A First Course in LInear Algebra' by Robert A. Beezer
36
Mathematics
Undergraduate 1
07/11/2014

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Term
Equation Operations Preserve Solution Sets (EOPSS)
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.

Term
Row-equivalent Matrices Represent Equivalent Systems (REMES)
Definition

Suppose A and B are row-equivalent augmented matrices. Then the systems of linear equations they represent are equivalent systems.

Term
Row Equivalent Matrix in Echelon Form (REMEF)
Definition

Suppose A is a matrix. Then there is a matrix B such that

  1. A and B are row-equivalent
  2. B is in row-reduced echelon form.
Term
Reduced Row Echelon Form is Unique (RREFU)
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.

Term
Recognizing Consistency of a Linear System (RCLS)
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.

 

 

 

Term
Consistent Systems, r and n (CSRN)
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.

Term
Free Variables for Consistent Systems (FVCS)
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.

Term
Possible Solution Sets for Linear Systems (PSSLS)
Definition

A system of linear equations has no solution, a unique solution or infinitely many solutions.

Term
Consistent, More Variables than Equations, Infinite (CMVEI)
Definition

Suppose a consistent system of linear equations has m equations in n variables. If n>m, then the system has infinitely many solutions.

Term
Homogeneous Systems Are Consistent (HSC)
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.

Term
Homogeneous, More Variables than Equations, Inifinite Solutions (HMVEI)
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.

Term
Nonsingular Matrices Row Reduce to the Identity Matrix (NMRRI)
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.

Term
Nonsingular Matrices Have Trivial Null Spaces (NMTNS)
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}

Term
Nonsingular Matrices And Unique Solutions (NMUS)
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.

Term
Nonsingular Matrix Equivalences (NME)
Definition

Suppose that A is a square matrix. The following are equivalent:

  1. A is nonsingular.
  2. A row-reduces to the identity matrix.
  3. The null space of A contains only the zero vector, N(A)={0}.
  4. The linear system LS(A,b) has a unique solution for every possible choice of b.
  5. The columns of A form a linearly independent set.
  6. A is invertible.
  7. The column space of A is [image], [image]
Term
Vector Space Properties of Column Vectors (VSPCV)
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]
Term
Solutions to Linear Systems are Linear Combinations (SLSLC)
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.]

Term
Vector Form of Solutions to Linear Systems (VFSLS)
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] ]

Term
Particular Solution Plus Homogeneous Solutions (PSPHS)
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].

Term
Spanning Sets for Null Spaces (SSNS)
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]

Term
Linear Independence Vectors and Homogeneous Systems (LIVHS)
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.

Term
Linearly Independent Vectors r and n (LIVRN)
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.

Term
More Vectors than Size Implies Linear Independence (MVSLD)
Definition

Suppose that [image] and n > m. Then S is a linearly independent set.

Term
Nonsingular Matrices have Linearly Independent Columns (NMLIC)
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.

Term
Basis for Null Spaces (BNS)
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

  1. N(A) = (S)
  2. S is a linearly independent set.
Term
Dependency in Linearly Dependent Sets (DLDS)
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].

Term
Basis of a Span (BS)
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

  1. [image] is a linearly independent set

  2. [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.)

Term
Conjugation Respects Vector Addition (CRVA)
Definition

Suppose x and y are two vectors from [image]. Then

 

[image]

 

Term
Conjugation Respects Vector Scalar Multiplication (CRSM)
Definition

Suppose x is a vector from [image], and [image] is a scalar. Then

 

[image]

 

Term
Inner Product and Vector Addition (IPVA)
Definition

Suppose [image]. Then

 

  1. [image]
  2. [image]

 

Term
Inner Product and Scalar Multiplication (IPSM)
Definition

Suppose [image]. Then

 

  1. [image]
  2. [image]

 

Term
Inner Product as Anti-Commutative (IPAC)
Definition

Suppose that u and v are vectors in [image]. Then

 

[image]

Term
Inner Products and Norms (IPN)
Definition

Suppose that u is a vector in [image]. Then

 

[image]

Term
Positive Inner Products (PIP)
Definition

Suppose that u is a vector in [image]. Then 

 

[image]

Term
Orthogonal Sets are Linearly Independent (OSLI)
Definition

Suppose that S is an orthogonal set of nonzero vectors. Then S is linearly independent.

Term
Gram-Schmidt Procedure (GSP)
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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