52 terms

Linear Equation

in variables x1...xn is an equation that can be written as a1x1+a2x2+...+anxn = b, where b and the coefficients a1...am are real or complex numbers.

System of linear equations

A collection of one or more linear equations involving the same variable

Solution

A list of numbers that makes each equation a true statement

Solution set

The set of all possible solutions

Equivalent linear systems

If they have the same solution set

A system of linear equations has

1) no solution

2) exactly one solution

3) infinitely many solutions

2) exactly one solution

3) infinitely many solutions

Consistent linear systems

If it has either one or infinitely many solutions

Inconsistent

If it has no solution

Matrix

A rectangular array

Coefficient matrix

An augmented matrix meaning a matrix containing variables that equal the last column of numbers

If the augmented matrices of two linear systems are row equivalent

Then the two systems have the same solution set

Echelon form

1) all nonzero rows are above any rows of all zeros

2) each leading entry of a row is in a column to the right of the leading entry above it

3)all entries in a column below a leading entry are zeros

2) each leading entry of a row is in a column to the right of the leading entry above it

3)all entries in a column below a leading entry are zeros

Reduced echelon form

1) contains the properties of echelon form

2) the leading entry in each nonzero row is 1

3) each leading 1 is the only nonzero entry in its column

2) the leading entry in each nonzero row is 1

3) each leading 1 is the only nonzero entry in its column

Uniqueness of the rref

Each matrix is row equivalent to one and only one reduced echelon matrix

Pivot position

A location in A that corresponds to a leading 1 in the rref of A. A pivot column is a column of A that contains a pivot position

Existence and unique theorem

Iff an echelon form of the augmented matrix does not contain a row in the form of [0 ... 0 b]

column vector (vector)

A matrix with only one column

Two vectors in R^2 are equal

Iff their corresponding entries are equal

Linear combination

Given vectors and scalars the vector y defined by y =c1v1 + c2v2 + ... + cnvn

The set of all linear combinations

Of v1 ... vp, if v1 ... vp are in R^n, is denoted by Span{v1, ..., vp} and is called the subset of Run spanned by v1 ... vp.

Ax

If A is an m

Linear Equation

in variables x1...xn is an equation that can be written as a1x1+a2x2+...+anxn = b, where b and the coefficients a1...am are real or complex numbers.

System of linear equations

A collection of one or more linear equations involving the same variable

Solution

A list of numbers that makes each equation a true statement

Solution set

The set of all possible solutions

Equivalent linear systems

If they have the same solution set

A system of linear equations has

1) no solution

2) exactly one solution

3) infinitely many solutions

2) exactly one solution

3) infinitely many solutions

Consistent linear systems

If it has either one or infinitely many solutions

Inconsistent

If it has no solution

Matrix

A rectangular array

Coefficient matrix

An augmented matrix meaning a matrix containing variables that equal the last column of numbers

If the augmented matrices of two linear systems are row equivalent

Then the two systems have the same solution set

Echelon form

1) all nonzero rows are above any rows of all zeros

2) each leading entry of a row is in a column to the right of the leading entry above it

3)all entries in a column below a leading entry are zeros

2) each leading entry of a row is in a column to the right of the leading entry above it

3)all entries in a column below a leading entry are zeros

Reduced echelon form

1) contains the properties of echelon form

2) the leading entry in each nonzero row is 1

3) each leading 1 is the only nonzero entry in its column

2) the leading entry in each nonzero row is 1

3) each leading 1 is the only nonzero entry in its column

Uniqueness of the rref

Each matrix is row equivalent to one and only one reduced echelon matrix

Pivot position

A location in A that corresponds to a leading 1 in the rref of A. A pivot column is a column of A that contains a pivot position

Existence and unique theorem

Iff an echelon form of the augmented matrix does not contain a row in the form of [0 ... 0 b]

column vector (vector)

A matrix with only one column

Two vectors in R^2 are equal

Iff their corresponding entries are equal

Linear combination

Given vectors and scalars the vector y defined by y =c1v1 + c2v2 + ... + cnvn

The set of all linear combinations

Of v1 ... vp, if v1 ... vp are in R^n, is denoted by Span{v1, ..., vp} and is called the subset of Run spanned by v1 ... vp.

Matrix equation

If A is an mxn matrix, with columns a1 ... an, and if x is in R^n, then the product of A and x is the linear combination of the columns of A using the corresponding entries in x as weights; x1a1 + x2a2 + ... + xnan

Theorem 3

If A is an mxn matrix, with columns a1...an, and if b is R^M, the matrix equation (Ax=b) has the same solution set as the vector equation (x1a1 + x2a2 + ... + xnan) which, in turn, has the same solution set as the system of linear equations whose augmented matrix is [a1 a2 .. an b]

Existence of solutions

The matrix equation has a solution iff if b is a linear combination of the columns of a

Theorem 4

Let A be an mxn matrix. All true or none at all.

1) For each b in R^m , the equation Ax=b has a solution

2) Each b in R^m is a linear combination of the columns of A

3) The columns of A span R^m

4) A has a pivot position in every row

1) For each b in R^m , the equation Ax=b has a solution

2) Each b in R^m is a linear combination of the columns of A

3) The columns of A span R^m

4) A has a pivot position in every row

homogeneous linear equation

Ax=0

Homogeneous equation contains a nontrivial solution

iff if the equation has at least on free variable

Linear Independence

An indexed set of vectors {v1, ..., vp} in R^n is said to be l.i. if the vector equation has only the trivial solution.

The columns of a matrix A are linearly independent

iff the homogeneous linear equation has only the trivial solution

A set of two vectors is linearly dependent if

at least one of the vectors is a multiple of the other

If a set contains more vectors than there are entries in each vector

then the set is linearly dependent. p > n

If a set S contains the zero vector,

then the set is linearly dependent