Linear algebra: definitions and theorems
1.1 SYSTEMS OF LINEAR EQUATIONS
A linear equation in the variables x 1 x n is an equation that can be written in the form a1x1 + a2x2 +
… + anxn = b where b and the coefficients a1, ... , an are real or complex numbers.
A system of linear equations (or a linear system) is a collection of one or more linear equations
involving the same variables.
A system of linear equations has 1. no solution, or 2. exactly one solution, or 3. infinitely many
solutions. A system of linear equations is said to be consistent if it has either one solution or infinitely
many solutions; a system is inconsistent if it has no solution.
ELEMENTARY ROW OPERATIONS 1. (Replacement) Replace one row by the sum of itself and a
multiple of another row. 2. (Interchange) Interchange two rows. 3. (Scaling) Multiply all entries in a
row by a nonzero constant.
Two matrices are called row equivalent if there is a sequence of elementary row operations that
transforms one matrix into the other. If the augmented matrices of two linear systems are row
equivalent, then the two systems have the same solution set.
1.2 ROW REDUCTION AND ECHELON FORMS
A leading entry of a row refers to the leftmost nonzero entry (in a nonzero row).
Uniqueness of the Reduced Echelon Form: Each matrix is row equivalent to one and only one
reduced echelon matrix.
A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon
form of A. A pivot column is a column of A that contains a pivot position.
1
, Basic variables: variables with only one value; free variables: you can choose any value for the x.
1.3 VECTOR EQUATIONS
A matrix with only one column is called a column vector, or simply a vector. Two vectors in R2 are
equal if and only if their corresponding entries are equal.
The vector whose entries are all zero is called the zero vector and is denoted by 0.
y = c 1v 1 C+ … + c p v p is called a linear combination of v 1 , … , vp with weights c1 , … , cp .
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1.1 SYSTEMS OF LINEAR EQUATIONS
A linear equation in the variables x 1 x n is an equation that can be written in the form a1x1 + a2x2 +
… + anxn = b where b and the coefficients a1, ... , an are real or complex numbers.
A system of linear equations (or a linear system) is a collection of one or more linear equations
involving the same variables.
A system of linear equations has 1. no solution, or 2. exactly one solution, or 3. infinitely many
solutions. A system of linear equations is said to be consistent if it has either one solution or infinitely
many solutions; a system is inconsistent if it has no solution.
ELEMENTARY ROW OPERATIONS 1. (Replacement) Replace one row by the sum of itself and a
multiple of another row. 2. (Interchange) Interchange two rows. 3. (Scaling) Multiply all entries in a
row by a nonzero constant.
Two matrices are called row equivalent if there is a sequence of elementary row operations that
transforms one matrix into the other. If the augmented matrices of two linear systems are row
equivalent, then the two systems have the same solution set.
1.2 ROW REDUCTION AND ECHELON FORMS
A leading entry of a row refers to the leftmost nonzero entry (in a nonzero row).
Uniqueness of the Reduced Echelon Form: Each matrix is row equivalent to one and only one
reduced echelon matrix.
A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon
form of A. A pivot column is a column of A that contains a pivot position.
1
, Basic variables: variables with only one value; free variables: you can choose any value for the x.
1.3 VECTOR EQUATIONS
A matrix with only one column is called a column vector, or simply a vector. Two vectors in R2 are
equal if and only if their corresponding entries are equal.
The vector whose entries are all zero is called the zero vector and is denoted by 0.
y = c 1v 1 C+ … + c p v p is called a linear combination of v 1 , … , vp with weights c1 , … , cp .
2
