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linear programming

feasible region

constraint

objective function

process of finding the minimum and maximum value of a functio…

the area of intersection of a system of inequalities

a limit or boundary of a region

the expression that defines the quantity to be maximized or m…

linear programming

process of finding the minimum and maximum value of a functio…

feasible region

the area of intersection of a system of inequalities

(i,j)-entry of A

The diagonal entries of A

Main diagonal

Diagonal matrix

The entry in A in the ith row and the jth column.

a11, a22,a33...

The diagonal of A formed by the diagonal entries.

A square n x n matrix whose nondiagonal entries are zero.

(i,j)-entry of A

The entry in A in the ith row and the jth column.

The diagonal entries of A

a11, a22,a33...

Y- intercept

Slope

Slope- intercept Form

Point- slope form

the y-coordinate of a point where a graph crosses the y-axis

the steepness of a line on a graph, equal to its vertical cha…

an equation written in the form y=mx+b is in slope-intercept…

y-y1 = m(x-x1), where m is the slope and (x1,y1) is the point…

Y- intercept

the y-coordinate of a point where a graph crosses the y-axis

Slope

the steepness of a line on a graph, equal to its vertical cha…

Vector Space

If a system is inconsistent, is there…

Span of a vector space (set of vectors…

Linearly dependent

A linear combination of v1...vn with coefficients c1...cn is…

No

The set of all possible linear combinations with vectors in t…

At least one vector in S can be written as a linear combinati…

Vector Space

A linear combination of v1...vn with coefficients c1...cn is…

If a system is inconsistent, is there…

No

Scatter Plot

Correlation

Trend Line

Line of Best Fit

a graph that relates two sets of data by plotting the data as…

relationship between data sets

a line that approximates the relationship between variables o…

a trend line that gives the most accurate model of related data

Scatter Plot

a graph that relates two sets of data by plotting the data as…

Correlation

relationship between data sets

The leading entry of the row is the:

Echelon Matrix Form

Reduced Echelon Matrix Form

Inconsistent/Consistent Matrices and E…

leftmost nonzero entry in that row

1. All nonzero rows are above any rows of zeros... 2. "Stairstep…

1. Same three reqs. as echelon form... 2. Leading entry is 1... 3.…

1. If there is a solution, there is a reduced echelon form to…

The leading entry of the row is the:

leftmost nonzero entry in that row

Echelon Matrix Form

1. All nonzero rows are above any rows of zeros... 2. "Stairstep…

system of linear inequalities

solutions of a system

half plane

Optimization

Graph of two linear inequalities using the same variables.

Any ordered pairs (x,y) that makes both inequalities true.

the part of a plane on one side of an indefinitely extended s…

A process in which the minimum or maximum of a quantity is so…

system of linear inequalities

Graph of two linear inequalities using the same variables.

solutions of a system

Any ordered pairs (x,y) that makes both inequalities true.

Definition of a subspace

Column Space

Null Space

Theorem 12 for subspaces

A subspace of R^n is any set H in R^n that has three properti…

The column space of a matrix A is the set Col A of all linear…

The null space of a matrix A is the set Nul A of all solution…

The null space of an mxn matrix A is a subspace of R^n. Equiv…

Definition of a subspace

A subspace of R^n is any set H in R^n that has three properti…

Column Space

The column space of a matrix A is the set Col A of all linear…

Linear Equation

System of Linear Equations

Solution of a System

Consistent System

In the variables x1, x2, x3, ...., xn is an equation that can…

A collection of one or more linear equations involving the sa…

a list (s1, s2, ..., sn) of numbers that makes each equation…

A system that has at least one solution

Linear Equation

In the variables x1, x2, x3, ...., xn is an equation that can…

System of Linear Equations

A collection of one or more linear equations involving the sa…

Determinant of n×n matrix.

(i,j)-cofactor of a matrix A.

Cofactor expansion across the first ro…

Determinant of a triangular matrix.

The number det A defined inductively by a cofactor expansion…

is the number Cij given by Cij = (-1)^(i+j) det Aij then det…

det A = a11C11 + a12C12 + ... + A1nC1n

the det A is the product of the entries on the main diagonal…

Determinant of n×n matrix.

The number det A defined inductively by a cofactor expansion…

(i,j)-cofactor of a matrix A.

is the number Cij given by Cij = (-1)^(i+j) det Aij then det…

For an m x n matrix, A, the rank of A…

What is the rank of A^T, the transpose…

True or False: If A is an m x n matrix…

What is the nullity of a 5 x 4 matrix…

The number of leading ones in reduced Echelon form.

The rank of A

True

N(A) = n - R(A) = 4 - 3 = 1

For an m x n matrix, A, the rank of A…

The number of leading ones in reduced Echelon form.

What is the rank of A^T, the transpose…

The rank of A

Diagonal Matrix

Matrix Addition

Addition Properties

When multiplying a matrix by a constant,

square nxn matrix with nonzero diagonal entries and zeros in…

add corresponding entries, matrices must be same size

a) A+B = B+A... b) (A+B)+C = A+(B+C)... c) A+0 = A... d) r(A+B) = rA+r…

all entries are multiplied by the constant

Diagonal Matrix

square nxn matrix with nonzero diagonal entries and zeros in…

Matrix Addition

add corresponding entries, matrices must be same size

Column Space

Null Space

Subspace

Basis for a Subspace

colA is the set of all possible linear combinations of column…

nulA is a set of solutions to Ax=0, Dependent columns

1. 0 vector is in H... 2. If u and v are in H, then u + v is in…

Any set of linearly independent vectors (not unique)

Column Space

colA is the set of all possible linear combinations of column…

Null Space

nulA is a set of solutions to Ax=0, Dependent columns

standard matrix

onto

one-to-one

zero matrix

For a linear transformation T:R^n->R^m, there exists a unique…

A transformation T:R^n->R^m is onto if every vector b in R^m…

A transformation T:R^n->R^m is one-to-one if every vector b i…

a matrix containing only zeros

standard matrix

For a linear transformation T:R^n->R^m, there exists a unique…

onto

A transformation T:R^n->R^m is onto if every vector b in R^m…

Similar matrices

Similarity transformation of matrices

Diagonalizable matrix

Diagonalization Theorem.

Matrices A and B such that P^-1AP = B or equivalently A = PBP…

it is when changing A into PDP^-1

A matrix that can be written in factored form as PDP^-1, wher…

An matrix A is diagonalizable if and only if A has n linearly…

Similar matrices

Matrices A and B such that P^-1AP = B or equivalently A = PBP…

Similarity transformation of matrices

it is when changing A into PDP^-1

Eigenvalues

eigenvectors

Diagonalization

Unit vector

a-j across diagonal, solve for zero

free variable in parametric form (basically null of eigenspace)

P is eigenvectors together, D is diagonal matrix with eigenva…

sqrt of sum of squares is one

Eigenvalues

a-j across diagonal, solve for zero

eigenvectors

free variable in parametric form (basically null of eigenspace)

ordered basis

coordinate vector of x relative to ord…

A*B, where A and B are mxn and nxp mat…

L_A

basis with specific order, for Rn it's {e_1, ... , e_n}, and…

[x]_beta = column version of (a1,a2,...,an)

(AB)_ij = (sum from k = 1 to n) A_ik * B_kj, 1 <= i <= m, 1 <…

L_A: Rn --> Rm, where L_A(x) = A*x

ordered basis

basis with specific order, for Rn it's {e_1, ... , e_n}, and…

coordinate vector of x relative to ord…

[x]_beta = column version of (a1,a2,...,an)

Partitioned (or block) matrix

Column--row expansion of AB

LU factorization

Solving the equation LUx=b

A matrix whose entries are themselves matrices of appropriate…

The expression of a product AB as a sum of outer products: co…

The representation of a matrix A in the form A = LU where L i…

L is the lower triangular form and U is the upper triangular…

Partitioned (or block) matrix

A matrix whose entries are themselves matrices of appropriate…

Column--row expansion of AB

The expression of a product AB as a sum of outer products: co…

linear programming

feasible region

constraint

objective function

process of finding the minimum and maximum value of a functio…

the area of intersection of a system of inequalities

a limit or boundary of a region

the expression that defines the quantity to be maximized or m…

linear programming

process of finding the minimum and maximum value of a functio…

feasible region

the area of intersection of a system of inequalities

A set of vectors is an orthogonal set…

If an orthogonal set consists of nonze…

An orthogonal basis for a subspace W o…

An m x m matrix U has orthonormal colu…

each pair of distinct vectors from the set is orthogonal

linearly independent and a basis for the subspace spanned by…

orthogonal set

U^TU = I

A set of vectors is an orthogonal set…

each pair of distinct vectors from the set is orthogonal

If an orthogonal set consists of nonze…

linearly independent and a basis for the subspace spanned by…

Formula

Solve an Equation

Equivalent Equations

Multi-Step Equations

A mathematical relationship or rule expressed in symbols.

The process of finding all values of the variable that make t…

equations that have the same solution

Equations with more than one operation.

Formula

A mathematical relationship or rule expressed in symbols.

Solve an Equation

The process of finding all values of the variable that make t…

T(u)=Au; nonsingular

lines; segments of lines; parallel lin…

||u|| = ||T(u)||; equal to; d(P, Q)=d(…

T(u)=u+v

A nonsingular transformation is a mapping defined by ________…

A nonsingular linear transformation T maps... (a) lines into ___…

Let T be an orthogonal transformation on R^n. Let u and v be…

A Translation is a transformation T:Rn... to Rm defined by _____…

T(u)=Au; nonsingular

A nonsingular transformation is a mapping defined by ________…

lines; segments of lines; parallel lin…

A nonsingular linear transformation T maps... (a) lines into ___…

Transformation T from Rⁿ to Rm

Linear Transformation

Theorem 10 (Ch. 1)

Standard matrix for the linear transfo…

a rule that assigns to each vector *x* in Rⁿ a vector T(*x*)…

A transformation is linear if ... (i) T(*u*+*v*) = T(*u*) + T(*v…

Let T: Rⁿ→Rm be a linear transformation. Then there exists a…

The matrix in A = [T(e₁) ... T(en)].

Transformation T from Rⁿ to Rm

a rule that assigns to each vector *x* in Rⁿ a vector T(*x*)…

Linear Transformation

A transformation is linear if ... (i) T(*u*+*v*) = T(*u*) + T(*v…

determinant properties (4)

if a has a zero column

if a has two equal columns

if one column of a is a multiple of an…

linearity, no change w/ column replacement, antisymmetry, norm

det(A)=0

det(A)=0

det(A)=0

determinant properties (4)

linearity, no change w/ column replacement, antisymmetry, norm

if a has a zero column

det(A)=0

Orthogonal projection

Properties of matrix with orthonormal…

Orthogonal matrix.

Orthogonal Decomposition Theorem.

y.u/u.u u

1. An mxn matrix U has orthonormal columns if and only if U^T…

a Square invertible matrix U such that U^-1 = U^T

^y = (y.u1/u.u) u1 + .... (y.up/up/up) up

Orthogonal projection

y.u/u.u u

Properties of matrix with orthonormal…

1. An mxn matrix U has orthonormal columns if and only if U^T…

Matrix Addition

Scalar Multiplication

Isomorphism

Standard Basis Vectors

Let A and B both be mxn matrices. Then the matrix A+B is the…

Let A be an mxn matrix and let c be a scalar. Then the matrix…

The behavior that when two elements of R^n are added as n-tup…

Vectors ei for i=1,...,n in R^n... Ex:... In R^1 we have one of the…

Matrix Addition

Let A and B both be mxn matrices. Then the matrix A+B is the…

Scalar Multiplication

Let A be an mxn matrix and let c be a scalar. Then the matrix…

A Square Matrix that has no inverse.

An n x n matrix A such that if there i…

An m x n matrix A, whose turned into a…

A square n x n matrix whose non diagon…

Singular (Matrix)

Invertible

Transpose

Diagonal Matrix

A Square Matrix that has no inverse.

Singular (Matrix)

An n x n matrix A such that if there i…

Invertible

Null Space

Column Space

Row Space

Left Null Space

the set of all vectors x that are solutions to the homogenous…

the set of all possible linear combinations of the matrix's c…

the column space the transpose of A; the set of all linear co…

the left null space is the null space of the transpose of A;…

Null Space

the set of all vectors x that are solutions to the homogenous…

Column Space

the set of all possible linear combinations of the matrix's c…

Define:... A Binary Operation

Define:... A Group

Define:... Abelian

Define:... Finite

A binary operation on a set G is a map G x G -> G from the Ca…

A group is a set G together with a binary operation G x G ->…

A group is called abelian if for every a and b in G, a.b = b.a

A group is called finite if it has finitely many elements.

Define:... A Binary Operation

A binary operation on a set G is a map G x G -> G from the Ca…

Define:... A Group

A group is a set G together with a binary operation G x G ->…

The Invertible Matrix Theorem

Let A and B be square matrices. If AB…

Let T be a linear transformation (R^n…

Column-Row expansion of AB

Let A be an nxn matrix. Then the following are equivalent:... a.…

A and B are both invertible, with B=A^-1 and A=B^-1

A is invertible

If A is mxn and B is nxp, then... AB=col1(A)row1(B)+...+coln(A)…

The Invertible Matrix Theorem

Let A be an nxn matrix. Then the following are equivalent:... a.…

Let A and B be square matrices. If AB…

A and B are both invertible, with B=A^-1 and A=B^-1

How to find if something is a vector s…

How to find if something is a subspace:

To find if a set of vectors are a span…

Row Echelon Form Requirements:

Follows the 8 axioms and is closed for scalar multiplication…

Show it is closed for scalar multiplication and addition:... α(x…

1. This means that a constant times each of vectors will equa…

1. The first entry in each row is 1... 2. For any row below it,…

How to find if something is a vector s…

Follows the 8 axioms and is closed for scalar multiplication…

How to find if something is a subspace:

Show it is closed for scalar multiplication and addition:... α(x…

If A and B are 2x2 matrices with colum…

Each column of AB is a linear combinat…

AB+AC=A(B+C)

A^T+B^T=(A+B)^T

False. Matrix multiplication is row by column

False. Swap A and B, then it's true.

True. Matrix multiplication distributes over addition

True. Properties of transportation. Also when we add we add c…

If A and B are 2x2 matrices with colum…

False. Matrix multiplication is row by column

Each column of AB is a linear combinat…

False. Swap A and B, then it's true.

Inconsistent

Consistent and independent

Consistent and dependent

Horizontal

If the slopes are the same but the y-intercepts are different

If the slopes are different

If the slopes are the same and the y-intercepts are the same

Y term

Inconsistent

If the slopes are the same but the y-intercepts are different

Consistent and independent

If the slopes are different

Leading entry

Rectangular matrix is in Row echelon f…

Rectangular matrix is in reduced Row e…

Row reduction

Leftmost nonzero entry in a row

1. All nonzero rows are above any rows of all zeroes... 2. Each…

In addition to the matrix being in echelon form:... 4. The lead…

Transformation of s matrix by elementary row operations

Leading entry

Leftmost nonzero entry in a row

Rectangular matrix is in Row echelon f…

1. All nonzero rows are above any rows of all zeroes... 2. Each…

Scalar Product, Dot Product, Inner Pro…

Dot product rules

Cauchy-Schwarz Inequality

Perpendicular

Legnth of vector x multiplied by the length of vector y multi…

Order doesnt matter, distributive accross addition, can take…

Length of the dot product of x and y is < length of x multipl…

Orthogonal, normal. 0 Dot product

Scalar Product, Dot Product, Inner Pro…

Legnth of vector x multiplied by the length of vector y multi…

Dot product rules

Order doesnt matter, distributive accross addition, can take…

solutions to systems of homogeneous eq…

span

How to determine if a vector is in a s…

Any 2 vectors that are NOT scalar mult…

subspaces

when a subspace of a vector space consists of all linear comb…

create a system of equations using the vector in question as…

will span all of R2

solutions to systems of homogeneous eq…

subspaces

span

when a subspace of a vector space consists of all linear comb…

Linear equation (In the variables x1..…

System of linear equations

Solution set of the linear system

Equivalent linear systems

An equation that can be written in the form a1x1 +a2x2+...anx…

A collection of one or more linear equations involving the sa…

A list of numbers(s1...sn) that makes each equation in the sy…

Linear systems with the same solution set.

Linear equation (In the variables x1..…

An equation that can be written in the form a1x1 +a2x2+...anx…

System of linear equations

A collection of one or more linear equations involving the sa…

Linear combination

Vector

Scalar

Parallelogram rule for addition

Doing both: adding two vectors and multiplying a scalar by a…

Matrix with only one column

Literally any real number: denoted as a vector with one entry

If u and v in R^2 are represented as points in the plane, the…

Linear combination

Doing both: adding two vectors and multiplying a scalar by a…

Vector

Matrix with only one column

Linear Independence

Linear Dependent

det|A|=0

det|A| doesn't equal 0

when the set of vectors has only the trivial solution

when the set of vectors have has solutions that are also nont…

there are infinite solutions, thus linearly dependent

only trivial solution exists, thus linearly independent

Linear Independence

when the set of vectors has only the trivial solution

Linear Dependent

when the set of vectors have has solutions that are also nont…

Definition 3.3.1: subspace

Definition 3.3.2: span

Theorem 3.3.3

Theorem 3.3.4

- If a collection of vectors V in Rn is also a vector space u…

- Let X be a collection of vectors in Rn. The span of X is th…

- If X is a collection of vectors in Rn, then Span(X) is a su…

- If V is a collection of vectors in Rn such that 0 is in V a…

Definition 3.3.1: subspace

- If a collection of vectors V in Rn is also a vector space u…

Definition 3.3.2: span

- Let X be a collection of vectors in Rn. The span of X is th…

Diagonalizable Matrices

Eigenvectors

Eigenvalue

Eigenbasis

Matrix A is diagonalizable iff A is similar to some diagonal…

A nonzero vector v in Rⁿ is ____ of A if Av = λv for some sca…

λ of the associated eigenvector. We find it by setting det(A-…

A basis formed from the eigenvectors for A, meaning that Av₁…

Diagonalizable Matrices

Matrix A is diagonalizable iff A is similar to some diagonal…

Eigenvectors

A nonzero vector v in Rⁿ is ____ of A if Av = λv for some sca…