# Study sets matching "algebra linear systems"

Study sets

Diagrams

Classes

Users

consistent

dependent

elimination

inconsistent

a system of linear equations that has at least one ordered pai…

a system of linear equations that has an infinite number of so…

the use of addition or subtraction to eliminate one variable a…

a system of equations with no ordered pair that satisfies both…

consistent

a system of linear equations that has at least one ordered pai…

dependent

a system of linear equations that has an infinite number of so…

system of two linear equations

ordered pair

exactly one solution

infinitely many solution

two variables x and y also called a linear system consists of…

(x,y), a point on a graph

system of two linear equations

two variables x and y also called a linear system consists of…

ordered pair

(x,y), a point on a graph

Function

Function Notation

Input

Output

A relationship where each input has exactly one output.

If x is input and y is the output, then the function notation…

A number in the domain of a function. ... The independent variabl…

A number in the range of a function.... The dependent variable.

Function

A relationship where each input has exactly one output.

Function Notation

If x is input and y is the output, then the function notation…

elimination method

linear inequality

solution of a system of linear inequali…

solution of a system of linear equations

solving systems by adding or subtracting equations to eliminat…

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 eliminat…

linear inequality

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

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,…-6

10

-1

3

Find the y-coordinate using elimination method: ... -4x - 2y = -1…

Find the x-coordinate using elimination method... x - y = 11... 2x +…

Find the y-coordinate using elimination method ... -6x + 5y = 1... 6…

Find the y-coordinate using elimination method... -2x - 9y = -25…

-6

Find the y-coordinate using elimination method: ... -4x - 2y = -1…

10

Find the x-coordinate using elimination method... x - y = 11... 2x +…

Common Solution

Elimination

Substitution

System of Linear Equation

x-y coordinate of the graphical intersection of the linear equ…

The process of solving a linear system of equations by adding…

The process of solving a linear system of equations when one v…

One or more linear equations together in a problem

Common Solution

x-y coordinate of the graphical intersection of the linear equ…

Elimination

The process of solving a linear system of equations by adding…

consistent system

dependent system

elimination method

inconsistent system

a system of equations or inequalities that has at least one so…

a system of equations that has infinitely many solutions

a method used to solve systems of equations in which one varia…

a system of equations or inequalities that has no solution

consistent system

a system of equations or inequalities that has at least one so…

dependent system

a system of equations that has infinitely many solutions

system of linear equations

solution of a system of linear equations

substitution method

system of linear inequalities

two or more linear equations with the same variables

the ordered pair that makes both equations true at the same ti…

one equation is solved for one of the variables then the the e…

two or more linear inequalities with the same variables

system of linear equations

two or more linear equations with the same variables

solution of a system of linear equations

the ordered pair that makes both equations true at the same ti…

Vector Space First Axiom

Vector Space Second Axiom

Vector Space Third Axiom

Vector Space Fourth Axiom

for all x,y under the set of V, x+y=y+x

for all x,y,z under the set of V, (x+y)+z=x+(y+z)

A zero vector exists in V, for all x in the set of V, x+(the z…

for all x under the set of V, the exists y under the set of V…

Vector Space First Axiom

for all x,y under the set of V, x+y=y+x

Vector Space Second Axiom

for all x,y,z under the set of V, (x+y)+z=x+(y+z)

Free Variable Theorem for Homogeneous s…

a homo geneous linear system with more…

If B and C are both inverses of the mat…

If A and B are invertible matrices with…

If a homogeneous linear system has n unknowns, and if the redu…

infinitley many solutions

since B is an inverse of A, we have BA=I. Multiplying both sid…

(AB)(B^-1A^-1)=(B^-1A^-1)(AB)=I , but ... (AB)(B^-1A^-1)=A(BB^-1)…

Free Variable Theorem for Homogeneous s…

If a homogeneous linear system has n unknowns, and if the redu…

a homo geneous linear system with more…

infinitley many solutions

consistent system

elimination

independent system

objective function

a system of equations or inequalities that has at least one so…

a method used to solve systems of equations in which one varia…

a system of equations that has exactly one solution

the function to be maximized or minimized in a linear programm…

consistent system

a system of equations or inequalities that has at least one so…

elimination

a method used to solve systems of equations in which one varia…

Rank

The Rank Theorem

Dimension Col(A)

Dimension Row(A)

The __________ of A is the dimension of the column space of A

Rank A + dim Nul(A) = n... {Number of pivot columns} + {Number of…

Number of pivot positions

Number of non-zero rows

Rank

The __________ of A is the dimension of the column space of A

The Rank Theorem

Rank A + dim Nul(A) = n... {Number of pivot columns} + {Number of…

system of equations

solution to a system of equations

parallel lines

right angles

a set of two or more equations that have the same variables

values that make both equations true

lines in the same plane that never intersect

angles with measure 90 degrees

system of equations

a set of two or more equations that have the same variables

solution to a system of equations

values that make both equations true

Theorem 4.7.2

Theorem 4.7.3

Theorem 4.7.4

Theorem 4.7.5

If x0 is any solution of a consistent linear system Ax=b, and…

Elementary row operation do not change the null space of a mat…

Elementary row operations do not change the row space of a mat…

if a matrix R is in row echelon form, then the row vectors wit…

Theorem 4.7.2

If x0 is any solution of a consistent linear system Ax=b, and…

Theorem 4.7.3

Elementary row operation do not change the null space of a mat…

non-singular

singular

echelon form

free

the matrix of coefficients of a homogeneous system with a uniq…

the matrix of coefficients of a homogeneous system with infini…

if each leading variable is to the right of the leading variab…

the variables that are not leading in an echelon form linear s…

non-singular

the matrix of coefficients of a homogeneous system with a uniq…

singular

the matrix of coefficients of a homogeneous system with infini…

There are only three types of solutions…

A system of linear equations is...

A system is called homogeneous if...

A system is called consistent if..

1. no solution. (parallel, no intersection, same slope)... 2. on…

is a list of linear equations in the same variables.

if each equation in the system is equal to zero. ... These system…

if there exists at least one solution. ... -a unique solution, wh…

There are only three types of solutions…

1. no solution. (parallel, no intersection, same slope)... 2. on…

A system of linear equations is...

is a list of linear equations in the same variables.

linear equation

slope

Types of Slope

y-intercept

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

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

Algebraically, the value of the output when the input is zero…

linear equation

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

slope

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

Solving if something is in a Span

Span Theorems

Span

Subset

By definition of span, we want to know if there are scalars x1…

Theorem 1: The subspace spanned by a non-empty subset S of a v…

The set of all linear combinations of the vectors.... or ... The s…

A subset is a set of vectors. Assume a subset V∈ℜn... Can be a s…

Solving if something is in a Span

By definition of span, we want to know if there are scalars x1…

Span Theorems

Theorem 1: The subspace spanned by a non-empty subset S of a v…

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,…Reduced row echelon form conditions

Theorem "dimension characterizes isomor…

Define "W is isomorphic to V"

Theorem stating "isomorphic to" is an e…

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 isom…

There exists an isomorphism T:W->V

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

Reduced row echelon form conditions

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

Theorem "dimension characterizes isomor…

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