Final Actually Flashcards
Linear Equation definition
An equation that can be written in the form a1x1+ a2x2+…+anxn =b
Coefficients
Real of complex numbers that are multiplied by x in the linear equation
A system of linear equations (linear system)
A collection of one or more linear equations involving the same variables.
The solution of a system
Is a list of all numbers that makes each equation a true statement when the values of s1,s2,…sn are substituted for x1, x2,… xn respectively.
Solution set
Set of all possible solutions
Equivalent
If they have the same solution set
How many solutions can a system of linear equations have
1) no solutions
2) one solution
3) infinite solutions
Consistent
Has one solution or infinitely many solutions
Inconsistent
No solutions
Coefficient matrix
A matrix of the coefficient of a set of linear equations, does not include answers
Augmented Matrix
The coefficient matrix with an added column containing the constants from the right sides of the equations.
What are the dimensions of an mxn matrix
The matrix has m rows and n columns
Three basic operations to solve linear systems
1) Replace one equation by the sum of itself and a multiple of another equation
2) Interchange two equtions
3) Multiply all terms in an equation by a nonzero constant
Row equivalent
Two matrices are row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other. They will have the same solution set
Two fundamental questions of linear systems:
1) Is the system consistent; that is, does at least one solution exist?
2) If a solution exists, is it the only one; that is, is the solution unique?
Nonzero row or column
A row or columns that contains at least one nonzero entry
Leading entry
Leftmost nonzero entry in a nonzero row
Row Echelon Form REF properties
- All nonzero rows are above any rows of all zeros
- Each leading entry of a row is in a column to the tight of the leading entry of the row above it
- All entries in a column below a leading entry are zeros
Reduce Row Echelon form properties
Same properties as REF plus:
4)The leading entry in each nonzero row is 1
5) Each leading 1 is the only nonzero entry in its column
(Is unique)
Pivot position
A location in A that corresponds to a leading 1 in the reduced echelon form RREF of A
Pivot column
Column that contains a pivot position
Basic variables
Variables of x corresponding to pivot columns.
Free variables
Variables not corresponding to a pivot column
Parameters
Free variables in a parametric equation
Column vector (vector)
A matrix with only one column
R^n
Denotes how many entries are in a column vector.
Scalar
A integer that is used to multiply a vector
Linear combination
A linear equation combined of scalars multplied by vectors
y = c1v1 + c2v2+…+cpvp
How are span and consistency related.
“is b in the span {…}?” Is the same as asking “is Ax=b consistent?”
Homogeneous
A system is homogeneous if it can be written in the form Ax=0
Trivial solution
The solution x = 0, which always exists for Ax=0
Linearly Independent
If the vector equation has only the trivial solution.
Transformation (or function or mapping)
A transformation T from R^n to R^m is a rule that assigns to each vector x in R^n a vector T(c) in R^m.
Domain
The set R^n is the domain of T
Codomain
R^m is the codomain of T
Range
The set of all images T(x) is the range of T
Superposition principle
T(c1v1+…+cpvp) = c1T(v1)+…+cpT(vp)
Contraction
Given a scalar r, define T: R^2 -> R^2 by T(x) =rx. When 0<=r<=1.
Dilation
Given a scalar r, define T: R^2 -> R^2 by T(x) =rx. When r> 1
Onto
1) A mapping T: R^n -> R^m is said to be onto R^m if eacb b in R^m is the image if at least one x in R^n
Or
2) T is onto R^m when the range of T is all the codomain of R^m
One to One
A mapping T: R^n -> R^m is said to be one to one if each b in R^m is the image of at most one x in R^n
Has either a unique solution or none at all
Diagonal Matrix
Is a square nxn matrix whose non diagonal entries are 0.
Invertible
An nxn matrix A is said to be invertible if there is an nxn matrix c such that CA=I and AC=I.
Singular Matrix
A matrix that is not invertible
Nonsingular matrix
An invertible matrix
Determinant of a 2x2 matrix
det(A) = ad-bc
Subspace
A subspace of R^n is any set H in R^n that has three properties:
1) The zero vector is in H
2) For each unit and v in H, the sum u+v is in H.
3) For each u in H and each scalar c, the vector cu is in H.
Zero subspace
Only contains the zero vector
Column Space
The column space of a matrix A is the set Col A of all linear combinations of the columns of A.
Null space
The null space of a matrix A is the set Nul A of all solutions of the homogeneous equation Ax=0
Basis
A basis for a subspace H of R^n is a linearly independent set in H that spans H
Dimension if a subspace
The dimension of a nonzero subspace H, denoted by dim H is the number of vectors in any basis for H. The dimension of the zero subspace {0} is defined to be zero.
Rank of a matrix
The rank of a matrix A, denoted by rank A, is the dimension of the column space of A.
(Or the number of pivot columns in A)
Rank of a matrix
The rank of a matrix A, denoted by rank A, is the dimension of the column space of A.
Determinant of a 2 by 2 matrix
Det A = ad-bc
Minor of entry a_ij
Denoted by M_ij and is defined to be the determinant of the sub matrix that remains after the ith row and the jth column are deleted from A
Cofactor entry of a_ij
The number (-1)^(i+j) M_ij is denoted by C_ij and is called the cofactor of entry a_ij
Eigenvector
An eigenvector of an nxn matrix A is a nonzero vector x such that Ax = lambda(x) for some scalar lambda.
Eigenvalue
A scalar lambda is called an eigenvalue of A if there is a nontrivial solution x of Ax=lambda(x) such that an x is called an eigenvector corresponding to lambda
Characteristic eqution
The equation det(A-lambda*I)=0
Diagonalizable
A square matrix is diagonalizable if A is similar to a diagonal matrix that is if A=PDP^-1
