# Study sets matching "linear math algebra"

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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

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…

Trivial Solution

Nontrivial Solution

Consistent System

Homogeneous System

The zero vector

An answer besides the zero vector. Exists if and only if there…

There is at least one solution to the system

System in which all the numbers on the right hand side are zer…

Trivial Solution

The zero vector

Nontrivial Solution

An answer besides the zero vector. Exists if and only if there…

Linear combination

Linear Equation

System of linear equations

Solution

an expression of the form c[1]A[1]+c[2]A[2]+...+c[n]A[n] where…

a linear combination of unknowns (or variables) set equal to a…

a finite set of linear equations that all use the same variabl…

an assignment of values to the variables that turns all of the…

Linear combination

an expression of the form c[1]A[1]+c[2]A[2]+...+c[n]A[n] where…

Linear Equation

a linear combination of unknowns (or variables) set equal to a…

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

Solution Set

Consistent System

Inconsistent System

Elementary Row Operations

What x and y equal after matrix has been reduced and solved

Solution Set contains a single point or infinitely many points…

Solution Set is empty

Replacement: Replace a row with the sum of itself and a multip…

Solution Set

What x and y equal after matrix has been reduced and solved

Consistent System

Solution Set contains a single point or infinitely many points…

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…

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

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 )

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

(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…

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…

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.

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

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"…

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…

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 subspa…

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

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,…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…