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invertible matrix

If A and B are invertible matrices of…

formula meaning A^n is invertible

If A is an invertible matrix, then

all A such that there exists A^-1 such that AA^-1 = A^-1A = I

AB is invertible, and (AB)^-1 = B^-1A^-1

(A^n)^-1 = (A^-1)^n

A^-1 is invertible, and A^n is invertible

invertible matrix

all A such that there exists A^-1 such that AA^-1 = A^-1A = I

If A and B are invertible matrices of…

AB is invertible, and (AB)^-1 = B^-1A^-1

Variables... Section 1-1 variables and ex…

Algebraic expressions... Section 1-1 vari…

Factors... Section 1-1 variables and expr…

Product... Section 1-1 variables and expr…

Are symbols used to represent unspecific numbers or values. A…

Consists of one or more numbers and variables along with one…

The quantities being multiplied... Example 2x4

The outcome of what is being multiplied... Example 2x4 product i…

Variables... Section 1-1 variables and ex…

Are symbols used to represent unspecific numbers or values. A…

Algebraic expressions... Section 1-1 vari…

Consists of one or more numbers and variables along with one…

Consistent System

Inconsistent System

Consistent Dependent System

Free variable

The matrix system has a solution

Matrix has no solution (last row 0 0 | non zero )

Infinite systems (last row 0 0 | 0 )

x1 x2 x3 etc can be any number, no set equation, no pivot in…

Consistent System

The matrix system has a solution

Inconsistent System

Matrix has no solution (last row 0 0 | non zero )

(VS1) ... Commutativity of Addition

(VS2)... Associativity of Addition

(VS3)... Zero Vector;... Identity Element of…

(VS4)... Inverse Elements of Addition

For all x, y in V, x+y = y+x

For all x, y, z in V (x+y)+z = x+ (y+z)

There exists an element in V denoted by 0 such that x+0=x for…

For each element x in V there exists and element y in V such…

(VS1) ... Commutativity of Addition

For all x, y in V, x+y = y+x

(VS2)... Associativity of Addition

For all x, y, z in V (x+y)+z = x+ (y+z)

Linear equation

Linear system (system of linear equati…

Solutions

Solution set

Equation that can be written in the form a₁x₁+a₂x₂+... = b

Collection of one or more linear equations involving the same…

Numbers that make each equation true

Set of all possible solutions

Linear equation

Equation that can be written in the form a₁x₁+a₂x₂+... = b

Linear system (system of linear equati…

Collection of one or more linear equations involving the same…

System of Linear Equations

Solution of a System of Linear Equations

Solution Set of a System of Linear Equ…

Equivalent Systems

A collection of m equations with the same variable quantities…

The solution of a system is a list (s1, s2, s3, ..., sn) of n…

The set of all possible solutions for a linear system.

When two linear systems have the SAME solution set.

System of Linear Equations

A collection of m equations with the same variable quantities…

Solution of a System of Linear Equations

The solution of a system is a list (s1, s2, s3, ..., sn) of n…

Thm 1

Thm 2

Linear Independence

Linear Dependence

A linear system has at least one solution if & only if the ri…

Let A be an mxn matrix coefficient matrix. Then the following…

The columns of A are linearly independent if & only if the tr…

The columns of A are linearly dependent if there is a nontriv…

Thm 1

A linear system has at least one solution if & only if the ri…

Thm 2

Let A be an mxn matrix coefficient matrix. Then the following…

Consistent System

Inconsistent System

Row Equivalent Matrices

Linear Combination

The system has either 1 solution or infinitely many

The system has no solutions

If a sequence of Elementary Row Operations can transform one…

Given vectors in R^n and scalars c1....cp, the vector b is gi…

Consistent System

The system has either 1 solution or infinitely many

Inconsistent System

The system has no solutions

Elementary row operations

Equivalent systems of linear equations

Matrix in row-echelon form properties

Matrix in reduced row-echelon form pro…

1. Interchange two rows... 2. Multiply a row by a nonzero consta…

Systems that have the same solution set

1. Rows of only zeros are at the bottom... 2. For each row that'…

It's in row-echelon form and every column that has a leading…

Elementary row operations

1. Interchange two rows... 2. Multiply a row by a nonzero consta…

Equivalent systems of linear equations

Systems that have the same solution set

Linear system

What are two basic methods of solving…

Two systems of linear equations are eq…

How do we know the method of eliminati…

A set of linear equations.

Method of elimination and substitution.

If they have the same solutions.

Each step in the method of elimination gives an equivalent sy…

Linear system

A set of linear equations.

What are two basic methods of solving…

Method of elimination and substitution.

x bar parallel is

two vectors are orthogonal to each oth…

magnitude is

In an LU decomposition, the columns of…

the orthogonal projection of x bar onto the line

u * v = 0

the square root of each component squared

the pivotal columns in original matrix

x bar parallel is

the orthogonal projection of x bar onto the line

two vectors are orthogonal to each oth…

u * v = 0

Linear equation

Coefficients

System of linear equations or Linear s…

Solution of the system

In the variables x(1)...x(n) is an equation that can be writt…

Real or complex numbers

Collection of one or more linear equations involving the same…

A list (s(1),s(2),...,s(n)) of numbers that makes each equati…

Linear equation

In the variables x(1)...x(n) is an equation that can be writt…

Coefficients

Real or complex numbers

Hilbert-Schmidt norm (for a matrix)

Matrix (operator) norm

Vandermonde matrix

Formula for the determinant of a Vande…

concatenate all the columns of the matrix end-to-end and take…

A measure of the maximum amount by which a matrix "stretches"…

Matrix used in Lagrangian interpolation (in 2D). Each row cor…

Plus or minus the product of the pairwise differences of ever…

Hilbert-Schmidt norm (for a matrix)

concatenate all the columns of the matrix end-to-end and take…

Matrix (operator) norm

A measure of the maximum amount by which a matrix "stretches"…

system of linear equations

linear equation

linear system

solution of a system

a collection of one ore more linear equations involving the s…

equation of first order with integer coefficients

same as linear equation

a list of numbers that makes each equation of a linear system…

system of linear equations

a collection of one ore more linear equations involving the s…

linear equation

equation of first order with integer coefficients

Vectors are linearly independent

Vector Space V is n-dimensional

Matrices can only be multiplied

Product cij

if the only solution of... a₁v₁ + ... + am vm = 0... is ai = 0, ∀i

dim V = n, if there exist linearly independent vectors, such…

if A is m x n and B is n x p

ⁿ∑ aik bkj

Vectors are linearly independent

if the only solution of... a₁v₁ + ... + am vm = 0... is ai = 0, ∀i

Vector Space V is n-dimensional

dim V = n, if there exist linearly independent vectors, such…

Reduced row echelon form conditions

Theorem "dimension characterizes isomo…

Define "W is isomorphic to V"

Theorem stating "isomorphic to" is an…

1. leading entry (if there is one) is 1 2. if column contains…

For subspaces V within R^n and W within R^m, we have V is iso…

There exists an isomorphism T:W->V

"isomorphic to" is an equivalence relation for subspaces V wi…

Reduced row echelon form conditions

1. leading entry (if there is one) is 1 2. if column contains…

Theorem "dimension characterizes isomo…

For subspaces V within R^n and W within R^m, we have V is iso…

Scalar multiplication

Standard Basis in Rn

Closed under addition

Closed under scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t…

Denoted by **e**n, [1, 0, 0, 0...0] to the nth number. So in R3…

Two vectors, when added together, are still in the same subsp…

When a vector is multiplied by a scalar, it's still in the sa…

Scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t…

Standard Basis in Rn

Denoted by **e**n, [1, 0, 0, 0...0] to the nth number. So in R3…

Scalar multiplication

Standard Basis in Rn

Closed under addition

Closed under scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t…

Denoted by **e**n, [1, 0, 0, 0...0] to the nth number. So in R3…

Two vectors, when added together, are still in the same subsp…

When a vector is multiplied by a scalar, it's still in the sa…

Scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t…

Standard Basis in Rn

Denoted by **e**n, [1, 0, 0, 0...0] to the nth number. So in R3…

What is a subspace and what are the re…

Column Space?

What is closure?

What is true for the span of a set of…

A subset (H) of a larger set of vectors(V) that has 3 propert…

denoted col(A)... the span of the columns of A.

When an operation such as addition is done with two numbers f…

For a set of vectors {v₁, v₂, v₃....} in Rⁿ... span{v₁,v₂,v₃...…

What is a subspace and what are the re…

A subset (H) of a larger set of vectors(V) that has 3 propert…

Column Space?

denoted col(A)... the span of the columns of A.

linear equation

consistent system

inconsistent system

leading entry

An equation that can be written as a1x1 + a2x2 + ... = b; a1,…

Has one or infinitely many solutions

Has no solution

Leftmost non-zero entry in a non-zero row

linear equation

An equation that can be written as a1x1 + a2x2 + ... = b; a1,…

consistent system

Has one or infinitely many solutions

Reduced row echelon form conditions

Theorem "dimension characterizes isomo…

Define "W is isomorphic to V"

Theorem stating "isomorphic to" is an…

1. leading entry (if there is one) is 1 2. if column contains…

For subspaces V within R^n and W within R^m, we have V is iso…

There exists an isomorphism T:W->V

"isomorphic to" is an equivalence relation for subspaces V wi…

Reduced row echelon form conditions

1. leading entry (if there is one) is 1 2. if column contains…

Theorem "dimension characterizes isomo…

For subspaces V within R^n and W within R^m, we have V is iso…

Vector Space

Vector Space Axioms

Subspace

Transpose

a set where addition and scalar multiplication as well as the…

1. Commutativity of addition... 2. Associativity in addition... 3.…

a subset of a vector space that is also a vector space closed…

Matrix formed from switching rows and colums (A₁₂)'=A₂₁

Vector Space

a set where addition and scalar multiplication as well as the…

Vector Space Axioms

1. Commutativity of addition... 2. Associativity in addition... 3.…

2 important observations regarding Mar…

Transition Matrix

If we can model an example as a Markov…

Markov Chain

1.) The market distribution at any specific time depends sole…

A matrix whose entries represent the probability of moving fr…

Sn = T^n * So... Sn = state of the system at time "n"... So = the…

A process in which the probability of a system being in a par…

2 important observations regarding Mar…

1.) The market distribution at any specific time depends sole…

Transition Matrix

A matrix whose entries represent the probability of moving fr…

3.1: a real number that is useful in t…

3.1 DEF 1: the _________ of a 2x2 matr…

3.1 DEF 2: if A is a square atrix, the…

3.1 DEF 3: If A is a ___________ of or…

-determinant

-determinant

-minor (Mij)... -determinant... -cofactor (Cij)... -Cij=(-1)^i+j(Mij)

-square matrix... -determinant... -inductive... -expanding by cofactor…

3.1: a real number that is useful in t…

-determinant

3.1 DEF 1: the _________ of a 2x2 matr…

-determinant

What is a subspace and what are the re…

Column Space?

What is closure?

What is true for the span of a set of…

A subset (H) of a larger set of vectors(V) that has 3 propert…

denoted col(A)... the span of the columns of A.

When an operation such as addition is done with two numbers f…

For a set of vectors {v₁, v₂, v₃....} in Rⁿ... span{v₁,v₂,v₃...…

What is a subspace and what are the re…

A subset (H) of a larger set of vectors(V) that has 3 propert…

Column Space?

denoted col(A)... the span of the columns of A.

What is a subspace and what are the re…

Column Space?

What is closure?

What is true for the span of a set of…

A subset (H) of a larger set of vectors(V) that has 3 propert…

denoted col(A)... the span of the columns of A.

When an operation such as addition is done with two numbers f…

For a set of vectors {v₁, v₂, v₃....} in Rⁿ... span{v₁,v₂,v₃...…

What is a subspace and what are the re…

A subset (H) of a larger set of vectors(V) that has 3 propert…

Column Space?

denoted col(A)... the span of the columns of A.

linear equality

linear inequality

half plane

system of linear inequalities

mathematical expression in which all variables appear to the…

linear expression containing an inequality sign rather than a…

those points (x,y) for which the inequality is true (all poin…

collection of more than one linear inequality

linear equality

mathematical expression in which all variables appear to the…

linear inequality

linear expression containing an inequality sign rather than a…

algebra

inconsistent system

matrix

square matrix

restoration (of broken parts)

a system with no solutions

rectangular array of numbers

a matrix in which the number of columns is equal to the numbe…

algebra

restoration (of broken parts)

inconsistent system

a system with no solutions

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…

Definition of a vector space

8 axioms of a vector space

Notation for a vector space with n dim…

Definition of a linear combination of…

V (set of elements: x, y, u, v) is called a vector space of K…

1) *u, v, w* ∈ V, (*u* + *v*) + *w* = *u* + (*v* + *w*) (Asso…

ℝⁿ

*x* is a linear combination of *v₁, ... vₙ* if for *v₁, ... v…

Definition of a vector space

V (set of elements: x, y, u, v) is called a vector space of K…

8 axioms of a vector space

1) *u, v, w* ∈ V, (*u* + *v*) + *w* = *u* + (*v* + *w*) (Asso…

Vectors provide what information?

Vector addition

Vector subtraction

Scalar multiplication

Magnitude (length) and direction

Done component-wise (is commutative*)... Vector v = <v1, v2, ..…

= one vector added to (-1)*another vector... Vector k = <-3, 2,…

Add the vector to itself however many times as indicated... For…

Vectors provide what information?

Magnitude (length) and direction

Vector addition

Done component-wise (is commutative*)... Vector v = <v1, v2, ..…

invariant subspace

eigenvalue

Equivalent conditions to be an eigenva…

The restriction operator

Suppose T in L(V). A subspace U of V is called invariant unde…

A number lambda in F is called an eigenvalue of T if... there ex…

(a) Lambda is an eigenvalue of T ;... (b) T - lambda I is not in…

T |U in L(U) is defined by... T |U (u) = Tu... for u in U

invariant subspace

Suppose T in L(V). A subspace U of V is called invariant unde…

eigenvalue

A number lambda in F is called an eigenvalue of T if... there ex…

In n variables; x1, x2, x3,....,xn has…

In in variables, is a set of m equatio…

1.1 DEF 1b: In this equation, a1, a2,.…

1.1 DEF 2a: A _______ of a linear equa…

linear equation

system of equations

-coefficients... -leading coefficients... -leading variable

-solution

In n variables; x1, x2, x3,....,xn has…

linear equation

In in variables, is a set of m equatio…

system of equations

What is a subspace and what are the re…

Column Space?

What is closure?

What is true for the span of a set of…

A subset (H) of a larger set of vectors(V) that has 3 propert…

denoted col(A)... the span of the columns of A.

When an operation such as addition is done with two numbers f…

For a set of vectors {v₁, v₂, v₃....} in Rⁿ... span{v₁,v₂,v₃...…

What is a subspace and what are the re…

A subset (H) of a larger set of vectors(V) that has 3 propert…

Column Space?

denoted col(A)... the span of the columns of A.

8 Conditions of vector space

Theorem 1.1: Cancellation Law for Vector

Corollary 1.1: Uniqueness of zero vector

Corollary 1.2: Uniqueness of additive…

P1: Closed under addition and scalar multiplication... 1. Additi…

Let V be vector space, if x, y, z are vectors in V such that…

The zero vector of a vector space is unique

The additive inverse of a vector in a vector space is unique

8 Conditions of vector space

P1: Closed under addition and scalar multiplication... 1. Additi…

Theorem 1.1: Cancellation Law for Vector

Let V be vector space, if x, y, z are vectors in V such that…

What is a vector?

What is a matrix?

What is the vector space?

What are the 8 axioms of vector space?

Vector: a quantity with more than one element (more than one…

Put simply, a rectangular array of numbers, symbols, expressi…

A vector space (also called a linear space) is a collection o…

What is a vector?

Vector: a quantity with more than one element (more than one…

What is a matrix?

Put simply, a rectangular array of numbers, symbols, expressi…

matrix notation for an entry

augmented matrix

row-echelon form

Gaussian elimination

(A)ij means the entry for A in the i row and the j column. Or…

where you have the solutions set up as another column; the ri…

Any rows entirely of 0 are at the bottom. Each row's first no…

*I think* where you do row operations starting at the top and…

matrix notation for an entry

(A)ij means the entry for A in the i row and the j column. Or…

augmented matrix

where you have the solutions set up as another column; the ri…

Algebraic equations

coefficient

consistent equation

constant

equation that includes one or more variables... 4x + 7 = 3x + 14

the number in front of the non-numerical symbols in an algebr…

an equation with only one equation

a term without a variable

Algebraic equations

equation that includes one or more variables... 4x + 7 = 3x + 14

coefficient

the number in front of the non-numerical symbols in an algebr…

A finite basis for V of Field F is a f…

Let S be a nonempty subset of a vector…

Let V & W be vector spaces over field…

A nonempty finite subset {u1, u2,.....…

Finite Basis

Span

Linear Transformations (Operators)

Linear Dependent

A finite basis for V of Field F is a f…

Finite Basis

Let S be a nonempty subset of a vector…

Span

Scalar multiplication

Standard Basis in Rn

Closed under addition

Closed under scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t…

Denoted by **e**n, [1, 0, 0, 0...0] to the nth number. So in R3…

Two vectors, when added together, are still in the same subsp…

When a vector is multiplied by a scalar, it's still in the sa…

Scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t…

Standard Basis in Rn

Denoted by **e**n, [1, 0, 0, 0...0] to the nth number. So in R3…

The number of solutions in a system of…

A system is said to be CONSISTENT if...

A system is said to be INCONSISTENT if…

Echelon Form

None, One, or Infinitely Many

it has one or infinitely many solutions

it has no solution

1- rows of zeroes on the bottom, 2- each leading entry of a r…

The number of solutions in a system of…

None, One, or Infinitely Many

A system is said to be CONSISTENT if...

it has one or infinitely many solutions

linear equation

system of linear equations

solution set

equivalent linear systems

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

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

a list of numbers that makes each equation in a system a true…

linear systems with the same solution set

linear equation

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

system of linear equations

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

Elementary Row Operations

Row Equivalency

Two equivalent equivalent augmented ma…

Two Fundamental Questions

1. Replacement... 2. Scaling... 3. Interchange

A is row equivalent to B if we can get from A to B via a sequ…

solution set

1. Is the system consistent... 2. If the solution exists, is the…

Elementary Row Operations

1. Replacement... 2. Scaling... 3. Interchange

Row Equivalency

A is row equivalent to B if we can get from A to B via a sequ…

Equation

Translating Words to Equations:... When…

1.) Understand the problem.... 2.) Choose…

Variable

is a mathematical statement that two expressions are equal. A…

1.) Read the word problem carefully to get an overview.... 2.) D…

EXAMPLE:... One-third of a number is fourteen.... 1/3 x n = 14... 1/3…

is a letter or symbol that represents an unknown quantity.

Equation

is a mathematical statement that two expressions are equal. A…

Translating Words to Equations:... When…

1.) Read the word problem carefully to get an overview.... 2.) D…