52 terms

Linear Algebra Final (Ch.1)

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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
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
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
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
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
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
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
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