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.

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

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.

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.

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.

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.

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.

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

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

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.

Suppose that a homogeneous system of linear equations has m equations and n variables with n > m. Then the system has infinitely many solutions.

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.

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}

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.

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]

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]

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 (A_j) which yield the constant (column) vector b.]

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] ]

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].

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]

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.

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.

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

Suppose that A is a square matrix. Then A is nonsingular if and only if the columns of A form a linearly independent set.

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.

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].

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.)

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

[image]

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

[image]

Suppose [image]. Then

1. [image]
2. [image]

Suppose [image]. Then

1. [image]
2. [image]

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

[image]

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

[image]

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

[image]

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

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].