#### Study sets matching "i linear algebra"

#### Study sets matching "i linear algebra"

linear equation

slope

y-intercept

x-intercept

An equation in two variables whose graph in a coordinate plan…

A ratio comparing the change in output over the change in inp…

The value of the output when the input is zero (0, y). The po…

The value of the input when the output is zero (x, 0). The po…

linear equation

An equation in two variables whose graph in a coordinate plan…

slope

A ratio comparing the change in output over the change in inp…

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

Quadrant

Solution of an equation in two variables

Graph of an equation in two variables

Linear equation

One of the four regions into which the x and y axes divide th…

An ordered pair that produces a true statement when the coord…

The set of points in a coordinate plane that represent all so…

An equation whose graph is a line.

Quadrant

One of the four regions into which the x and y axes divide th…

Solution of an equation in two variables

An ordered pair that produces a true statement when the coord…

Equivalent Systems

consistent system

inconsistent system

Row Echelon Form Qualifications [REF]…

Have the same solution set

either one solution or infinitely many

no solution

1. All zero rows below nonzero rows... 2. Each leading entry of…

Equivalent Systems

Have the same solution set

consistent system

either one solution or infinitely many

Linear Equation

Coefficients

System of Linear Equations/Linear System

Solution

a1x1 + a2x2 +... + anxn=b

a1...an are real or complex numbers.

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

A list of numbers that makes each equation a true statement w…

Linear Equation

a1x1 + a2x2 +... + anxn=b

Coefficients

a1...an are real or complex numbers.

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.…

Matrix equation

Theorem 3

Identity matrix

Theorem 4

an equation that involves at least one matrix for instance, A…

If A is an m*n matrix, with columns a1...an, and if b is in R…

A square matrix with ones on the diagonal and zeros elsewhere

Let A be an m*n matrix. Then the following statements are log…

Matrix equation

an equation that involves at least one matrix for instance, A…

Theorem 3

If A is an m*n matrix, with columns a1...an, and if b is in R…

basic variable

existance

consistent

inconsistant

variable that corresponds to a pivot column in the coefficien…

there is at least one solution to the matrix

the system of equations has at least one solution

the system has no solutions

basic variable

variable that corresponds to a pivot column in the coefficien…

existance

there is at least one solution to the matrix

Slope

Y-intercept

Standard Form

Slope-intercept Form

(y₂-y₁)/(x₂-x₁)

The point where the graph crosses the y-axis. In a linear equ…

Ax + By=C, where A, B, and C are not decimals or fractions, w…

y=mx+b, where m is the slope and b is the y-intercept of the…

Slope

(y₂-y₁)/(x₂-x₁)

Y-intercept

The point where the graph crosses the y-axis. In a linear equ…

solution

Gauss-Jordan elimination

elementary transformations

Echelon system

pair of x and y values that satisfies all equations in a syst…

1) put in augmented matrix form... 2) make triangle of zeros on…

interchanging equations, multiplying a row by a nonzero const…

3 zeros in the lower left hand corner

solution

pair of x and y values that satisfies all equations in a syst…

Gauss-Jordan elimination

1) put in augmented matrix form... 2) make triangle of zeros on…

X-Intercept

Y-Intercept

Linear Function

Rate

The x-coordinate(s) of the point(s) where a graph intersects…

The y-coordinate(s) of the point(s) where a graph intersects…

A function that can be written in the form y=mx+b, where x is…

A ratio that compares two quantities measured in different un…

X-Intercept

The x-coordinate(s) of the point(s) where a graph intersects…

Y-Intercept

The y-coordinate(s) of the point(s) where a graph intersects…

linear equation

slope

y-intercept

x-intercept

An equation in two variables whose graph in a coordinate plan…

A ratio comparing the change in output over the change in inp…

The value of the output when the input is zero (0, y). The po…

The value of the input when the output is zero (x, 0). The po…

linear equation

An equation in two variables whose graph in a coordinate plan…

slope

A ratio comparing the change in output over the change in inp…

coordinate plane

origin

x-axis

y-axis

the grid created by the horizontal x-axis and the vertical y-…

(0,0) in the coordinate plane; center of the grid

the horizontal number line in the coordinate plane (left to r…

the vertical number line in the coordinate plane (up and down)

coordinate plane

the grid created by the horizontal x-axis and the vertical y-…

origin

(0,0) in the coordinate plane; center of the grid

rate or rate of change

constant rate of change

constant rate of change

zero rate

measure like miles per hour or price per item

rate that stays the same throughout the table

graph of this rate is a straight line

graph of this rate is a horizontal line

rate or rate of change

measure like miles per hour or price per item

constant rate of change

rate that stays the same throughout the table

Linear Equation

Linear Coefficients

System of linear equations (linear sys…

Solution of a linear system

1. variables x1......xn that can be written in the form:... a1x1…

Can be real or complex numbers

Collection of one of more linear equations that contain x1...…

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

Linear Equation

1. variables x1......xn that can be written in the form:... a1x1…

Linear Coefficients

Can be real or complex numbers

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…

Homogeneous Linear System

Trivial/Nontrivial Solutions

The homogeneous equation Ax=0 has a no…

A nontrivial solution x can have some…

1. A system of linear equations is said to be homogeneous if…

1. The zero solution is called the trivial solution... 2. The no…

the equation has at least one free variable.

not all of the entries are zero.

Homogeneous Linear System

1. A system of linear equations is said to be homogeneous if…

Trivial/Nontrivial Solutions

1. The zero solution is called the trivial solution... 2. The no…

Is the statement "Every elementary ro…

Is the statement "A 5×6 matrix has si…

Is the statement "Two fundamental que…

Is the statement "Two matrices are ro…

True, because replacement, interchanging, and scaling are al…

False, because a 5×6 matrix has five rows and six columns.

True, because two fundamental questions address whether the s…

False, because if two matrices are row equivalent it means th…

Is the statement "Every elementary ro…

True, because replacement, interchanging, and scaling are al…

Is the statement "A 5×6 matrix has si…

False, because a 5×6 matrix has five rows and six columns.

(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...

The number c in cu is a _________.

R^2 is a _______.

Vectors in R^3 are ______ X _______ co…

Vectors in R^3 are represented in ____…

scalar

rectangular coordinate system in a plane.

3; 1

three

The number c in cu is a _________.

scalar

R^2 is a _______.

rectangular coordinate system in a plane.

Linearly independent

Linearly dependent

Linear Independence of Matrix Columns

Set of One Vector

1. Indexed set of vectors {v1,...vp} in R(n) is said to be li…

1. Indexed set of vectors {v1,...vp} in R(n) is said to be li…

The columns of a matrix A are linearly independent iff the eq…

1. A set of one vector is linearly independent iff v is NOT t…

Linearly independent

1. Indexed set of vectors {v1,...vp} in R(n) is said to be li…

Linearly dependent

1. Indexed set of vectors {v1,...vp} in R(n) is said to be li…

Vector

R^(n)

R2

Equal R2 vectors

An ordered list of members

-R stands for the real numbers that appear as entries in the…

The set of all vectors with two entries

Two vectors in R2 are equal iff their corresponding entries a…

Vector

An ordered list of members

R^(n)

-R stands for the real numbers that appear as entries in the…

Definition of Linear Combination

Matrix Equation

Theorem 3

Solution of Ax=b

1. If A is an m x n matrix, with columns a1, ... an, and if x…

Ax = b

1. If A is an m x n matrix with columns a1,...,an, and if b i…

1. The equation Ax=b has a solution iff b is a linear combina…

Definition of Linear Combination

1. If A is an m x n matrix, with columns a1, ... an, and if x…

Matrix Equation

Ax = b

If the augmented matrices of 2 linear…

Another way to ask if a system is cons…

Echelon Form (properties)

Reduced Echelon Form (properties)

the 2 have the same solution set

does at least 1 solution exist

All non zero rows above zero rows... each leading entry of a ro…

All properties from Echelon Form... leading entry is 1 in each…

If the augmented matrices of 2 linear…

the 2 have the same solution set

Another way to ask if a system is cons…

does at least 1 solution exist

consistent linear system

inconsistent linear system

free variable

vector

a linear system with at least one solution

a linear system with no solution

any variable in a linear system that is not a basic variable

a list of numbers; a matrix with only one column

consistent linear system

a linear system with at least one solution

inconsistent linear system

a linear system with no solution

Linearly independent and linearly depe…

Linear dependence relation

The columns of matrix A are linearly i…

A set containing only one vector is li…

the equation A

**x**=**0**has only the trivial solution.that vector is not the zero vector.

Linearly independent and linearly depe…

Linear dependence relation

Linear Combination

Can b be written as a linear combinati…

Span{}

Vector Equations

Given a set of vectors v1, v2, v3,...vp in Rn, and scalars c1…

The vector equation c1

**v1 + c2**v2 + ... cp*vp = b can be writ…The set of ALL linear combinations of the vectors.

c1

**v1 + c2**v2 + ... cp*vp = bLinear Combination

Given a set of vectors v1, v2, v3,...vp in Rn, and scalars c1…

Can b be written as a linear combinati…

The vector equation c1

**v1 + c2**v2 + ... cp*vp = b can be writ…Vector

Length of Vector

Unit Vector

scalar multiple

A real n-vector is an ordered sequence on n real #s. The set…

||a||=√a1^2+...+an^2

vector of length 1

let x=[x1...xn] be a vector in rn and let c be any scalar. th…

Vector

A real n-vector is an ordered sequence on n real #s. The set…

Length of Vector

||a||=√a1^2+...+an^2

Consistent

Inconsistent

linear equation

system of linear equations

Either has one solution or infinitely many solutions

no solutions

in the variables x1...xn is an equation that can be written i…

collection of one or more linear equations involving the same…

Consistent

Either has one solution or infinitely many solutions

Inconsistent

no solutions

Slope

Y-intercept

Standard Form

Slope-intercept Form

(y₂-y₁)/(x₂-x₁)

The point where the graph crosses the y-axis. In a linear equ…

Ax + By=C, where A, B, and C are not decimals or fractions, w…

y=mx+b, where m is the slope and b is the y-intercept of the…

Slope

(y₂-y₁)/(x₂-x₁)

Y-intercept

The point where the graph crosses the y-axis. In a linear equ…

linear equation

slope

y-intercept

x-intercept

An equation in two variables whose graph in a coordinate plan…

A ratio comparing the change in output over the change in inp…

The value of the output when the input is zero (0, y). The po…

The value of the input when the output is zero (x, 0). The po…

linear equation

An equation in two variables whose graph in a coordinate plan…

slope

A ratio comparing the change in output over the change in inp…

A linear equation is...

A system of linear equations (a linear…

A solution to the linear system x_1, .…

Solving a linear system means...

variables x_1, x_2,...,x_n, any equation that can be written…

x_1,x_2,...,x_n a collection of linear equations that is the…

any ordered n-tuple (S_1,...,S_n) of real numbers if you subs…

to find the solution set, i.e. the set of all possible soluti…

A linear equation is...

variables x_1, x_2,...,x_n, any equation that can be written…

A system of linear equations (a linear…

x_1,x_2,...,x_n a collection of linear equations that is the…

linear equation

system of linear equations

solution

solution set

a1x1+a2x2+...+anxn=b

collection of linear equations

a list of numbers that make each equation a true statement

a set of all possible solutions

linear equation

a1x1+a2x2+...+anxn=b

system of linear equations

collection of linear equations

vector space

subspaces

A + B

λA

Let F be a field (R or C). A vector space over F is a set V w…

Let V be a vector space. A subspace of V is a non-empty subse…

A + B := {v ∈ V | ∃a ∈ A, ∃b ∈ B : v = a + b}

λA := {v ∈ V | ∃a ∈ A : v = λa}

vector space

Let F be a field (R or C). A vector space over F is a set V w…

subspaces

Let V be a vector space. A subspace of V is a non-empty subse…

elimination method

linear inequality

solution of a system of linear inequal…

solution of a system of linear equations

solving systems by adding or subtracting equations to elimina…

an inequality in two variables whose graph is a region of the…

a set of values that make all inequalities in a system true.…

a set of values that make all equations in a system true. [p.…

elimination method

solving systems by adding or subtracting equations to elimina…

linear inequality

an inequality in two variables whose graph is a region of the…

Transformation/function/mapping /T

Domain

Codomain

Range

A rule that assigns to each vector

**x**in R^n to a vector in…The vector space R^n on which a transformation acts.

The vector space R^m that contains (but is no necessarily exh…

The set of all possible images T(

**x**) of transformation T.Transformation/function/mapping /T

A rule that assigns to each vector

**x**in R^n to a vector in…Domain

The vector space R^n on which a transformation acts.

Every linear transformation is...

Let T: R^n -> R^m be a linear transfor…

Standard matrix for the linear transfo…

Standard matrix for the linear transfo…

a matrix transformation (within finite dimensional spaces).

standard basis

Every linear transformation is...

a matrix transformation (within finite dimensional spaces).

Let T: R^n -> R^m be a linear transfor…

coefficient

distributive property

equation

inequality

the numerical part of a term

for any numbers a, b, and c: a(b+c) = ab+bc

a mathematical sentence showing two expressions are equal

mathematical sentences having the symbols <,>

coefficient

the numerical part of a term

distributive property

for any numbers a, b, and c: a(b+c) = ab+bc