# Study sets matching "linear math algebra"

invertible matrix

If A and B are invertible matrices of t…

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 t…

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

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

Algebraic expressions... Section 1-1 varia…

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

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

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

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

The quantities being multiplied... Example 2x4

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

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

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

Algebraic expressions... Section 1-1 varia…

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

equivalent system of equations

A Homogeneous system

invertible matrix

singularity

when two system of equations have the same solution set

when the constants on the right hand side of the matrix are al…

when there exists a matrix B such that AB=BA=I.

when a nxn matrix does not have a multiplicative inverse.

equivalent system of equations

when two system of equations have the same solution set

A Homogeneous system

when the constants on the right hand side of the matrix are al…

Consistent System

Inconsistent System

Properties of Matrix Addition and Scala…

Properties of Zero Matricies

A system of equations that has at least one solution

A system of equations that has no solution

1. A+B = B+A... 2. A+(B+C) = (A+B)+C... 3. (cd)A = c(dA)... 4. IA = A... 5…

1. A+0 = A... 2. A+(-A) = 0... 3. If cA = 0, then c=0 or A=0

Consistent System

A system of equations that has at least one solution

Inconsistent System

A system of equations that has no solution

linear equation

system of linear equations (linear syst…

solution

A system of linear equations has

an equation that can be written in the form ax+ax+ax+...+ax=b…

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

A solution of the system is a list (s,s,s,s..,s) of numbers th…

1. no solution... 2. exactly one solution... 3. infinitely many solu…

linear equation

an equation that can be written in the form ax+ax+ax+...+ax=b…

system of linear equations (linear syst…

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

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 t…

Consistent System

The matrix system has a solution

Inconsistent System

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

Thm 1

Thm 2

Linear Independence

Linear Dependence

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

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

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

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

Thm 1

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

Thm 2

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

(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 t…

(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)

System of Linear Equations

Solution of a System of Linear Equations

Solution Set of a System of Linear Equa…

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 nu…

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 nu…

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 i…

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

Consistent System

The system has either 1 solution or infinitely many

Inconsistent System

The system has no solutions

Thm. 4.2.1 - If B is produced from A by…

Thm. 4.2.1 - If B is produced from A by…

Thm. 4.2.1 - If B is produced from A by…

Thm. 4.2.2 - If A has two identical row…

det(B) = kdet(A)

det(B) = -det(A)

det(B) = det(A)

det(A) = 0

Thm. 4.2.1 - If B is produced from A by…

det(B) = kdet(A)

Thm. 4.2.1 - If B is produced from A by…

det(B) = -det(A)

Linear system

What are two basic methods of solving s…

Two systems of linear equations are equ…

How do we know the method of eliminatio…

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 sys…

Linear system

A set of linear equations.

What are two basic methods of solving s…

Method of elimination and substitution.

Linear equation

Linear system (system of linear equatio…

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 equatio…

Collection of one or more linear equations involving the same…

Elementary row operations

Equivalent systems of linear equations

Matrix in row-echelon form properties

Matrix in reduced row-echelon form prop…

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

Systems that have the same solution set

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

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

Elementary row operations

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

Equivalent systems of linear equations

Systems that have the same solution set

x bar parallel is

two vectors are orthogonal to each othe…

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 othe…

u * v = 0

vector space X over 𝐹

subspace

linear combination

span of a subset of vectors .

is a nonempty set with two operations, "addition" and "scalar…

Vector spaces may be formed from subsets of other vectors spac…

one vector is equal to the sum of scalar multiples of other ve…

is the set of all possible linear combinations of u1,u2,u3,...…

vector space X over 𝐹

is a nonempty set with two operations, "addition" and "scalar…

subspace

Vector spaces may be formed from subsets of other vectors spac…

Hilbert-Schmidt norm (for a matrix) (ak…

Matrix (operator) norm

Vandermonde matrix

Formula for the determinant of a Vander…

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 corr…

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

Hilbert-Schmidt norm (for a matrix) (ak…

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"…

1.1.1 Linear Combination

1.1.2 Linear Equation

1.1.3 System of Linear equations

1.1.4 Solution

...of A_1, A_2, A_3...expression of the form c_1A_1+...c_nA_n

>Linear combination of unknowns (variables) set equal to a con…

Finite set of linear equations that all use the same variables…

>Assignment of numbers to each unknown that makes the identity…

1.1.1 Linear Combination

...of A_1, A_2, A_3...expression of the form c_1A_1+...c_nA_n

1.1.2 Linear Equation

>Linear combination of unknowns (variables) set equal to a con…

Vector Space

Subspace

Linear Transformation

Source & Target

A vector space is a set V , equipped with a rule for addition…

If V is a vector space, a subspace of V is a non-empty subset…

Let V and W be vector spaces. A linear transformation is a map…

Suppose V −→ W is a linear transformation. The vector space V…

Vector Space

A vector space is a set V , equipped with a rule for addition…

Subspace

If V is a vector space, a subspace of V is a non-empty subset…

Linear equation

Coefficients

System of linear equations or Linear sy…

Solution of the system

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

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 equatio…

Linear equation

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

Coefficients

Real or complex numbers

system of linear equations

linear equation

linear system

solution of a system

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

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 sa…

linear equation

equation of first order with integer coefficients