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Study sets matching "1 linear algebra"

31 terms
Algebra 1 Linear Review
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31 terms
Algebra 1 Linear Review
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17 terms
Algebra 1: Linear Vocabulary
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…
31 terms
Algebra 1 Linear Review
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31 terms
Algebra 1 Linear Review
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32 terms
Linear Algebra Test 1
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.…
31 terms
Algebra 1: Linear Review
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14 terms
Algebra 1: Linear Equations
Rate of change
Slope
The Slope Formula
Positive Slope
Rate of change
Slope
Linear Algebra Chapter 1
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
21 terms
Algebra 1 Linear Equations
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26 terms
2.1 Linear Algebra
(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...
25 terms
Linear Functions (Algebra 1)
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…
20 terms
Algebra 1: Vocabulary (Linear, Slope)
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
41 terms
Linear Algebra Exam 1
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
104 terms
Linear Algebra Chapter 1
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.
18 terms
Linear Algebra Test 1
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
10 terms
Linear Algebra 1.7
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 Ax = 0 has only the trivial solution.
that vector is not the zero vector.
Linearly independent and linearly depe…
Linear dependence relation
14 terms
Algebra 1: Linear (w/ pictures)
Rate of change
Slope
The Slope Formula
Positive Slope
Rate of change
Slope
64 terms
Algebra 1: Linear and Inequality Review
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Linear Algebra 1:1
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
7 terms
Algebra 1: Systems of Linear Equations
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…
11 terms
Linear Algebra 1:2
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…
13 terms
Linear Algebra 1:5
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…
15 terms
Algebra 1 - Linear Equations and Inequalities
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
24 terms
Linear Algebra 1.9
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…
10 terms
Linear Algebra 1.8
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.
32 terms
Linear Algebra1
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.…
29 terms
Linear Algebra Exam #1
1 parameter
2 parameters
3 parameters
mxn
line
plane
hyper-plane
m rows by n columns
1 parameter
line
2 parameters
plane
20 terms
Algebra 1 Unit 4: Linear Functions
75 kilometers
300 kilometers
0 < y < 350
The original line went uphill and the…
A car travels at the rate shown in the graph. How far will it…
How far will the car travel in 6 hours?
Let f(x) = 5x where x represents speed in miles per hour of a…
What happens to the line represented by the equation, ... y = 4…
75 kilometers
A car travels at the rate shown in the graph. How far will it…
300 kilometers
How far will the car travel in 6 hours?
14 terms
Linear Algebra Section 1.1
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
10 terms
Algebra 1- Linear Equations
Standard Form
Slope Intercept Form
Vertical line
Horizontal line
ax+by=c
y=mx+b
x=a m=undefined
y=b m=0
Standard Form
ax+by=c
Slope Intercept Form
y=mx+b
15 terms
Linear Algebra Chapter 1
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
66 terms
Linear Algebra Test 1
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
Linear Algebra 1.3
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.
9 terms
Linear Algebra 1:8
Transformation
Domain of T
Codomain of T
Image
A transformation T from R(n) -> R(m) is a rule that assigns t…
Set R(n)
Set R(m)
For x in R(n), the vector T(x) in R(m) is called the image of…
Transformation
A transformation T from R(n) -> R(m) is a rule that assigns t…
Domain of T
Set R(n)
22 terms
Linear Algebra - Chapter 1
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 c1v1 + c2v2 + ... cp*vp = b can be writ…
The set of ALL linear combinations of the vectors.
c1v1 + c2v2 + ... cp*vp = b
Linear 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 c1v1 + c2v2 + ... cp*vp = b can be writ…
11 terms
Linear Algebra 1:7
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…
16 terms
Linear Algebra 1:3
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…
11 terms
Linear Algebra 1:4
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
44 terms
Linear Algebra Chapter 1
Vector
Problem with definitions (6th Grade Ex…
Infusing Tech into Math Article
Velocity
a. Math. A quantity having direction as well as magnitude, de…
The truth is that all definitions, by their very nature, tend…
1) the student teachers' attitudes about using technology in…
Knowing that a ship has a speed of 10 knots (nautical miles p…
Vector
a. Math. A quantity having direction as well as magnitude, de…
Problem with definitions (6th Grade Ex…
The truth is that all definitions, by their very nature, tend…
45 terms
Linear Algebra T1
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
11 terms
1.4 Linear Algebra
Matrix equation
The matrix equation Ax = b has the…
The equation Ax = b only has a solut…
"The columns of A span R^m" means that…
the vector equation x1a1 + x2a2 + ... xnan = b if A is…
b is a linear combination of the columns of A, which is equ…
every vector in R^m is a linear combination of v1 ... vp
Matrix equation
The matrix equation Ax = b has the…
the vector equation x1a1 + x2a2 + ... xnan = b if A is…
26 terms
Linear Algebra Exam 1
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
13 terms
linear algebra exam 1
linear equation
system of linear equations
solution of a system
consistent system
in the variables x₁,x₂,x₃..... is an equation that can be wri…
is a collection of linear equations involving the same variab…
of linear equations is a list of numbers that is a solution o…
one or infinitely solutions
linear equation
in the variables x₁,x₂,x₃..... is an equation that can be wri…
system of linear equations
is a collection of linear equations involving the same variab…
55 terms
Introduction to Linear Algebra 1
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…
17 terms
Linear Algebra 1.2
Reduced Echelon Form
Echelon Form
Neither
...
...
...
Reduced Echelon Form
...
Echelon Form
...
28 terms
Linear Algebra Test 1
Row-Reduced Echelon Form
Row Equivalence
Equivalence Relation
Partition
1. Must be in Row Echelon Form... 2. Every lead coefficient is 1…
Two matrices, A and B, are row equivalent if it is possible t…
A binary relation on a set that must be reflexive, symmetric,…
A grouping of a set's elements that follows 3 rules:... 1. does…
Row-Reduced Echelon Form
1. Must be in Row Echelon Form... 2. Every lead coefficient is 1…
Row Equivalence
Two matrices, A and B, are row equivalent if it is possible t…
20 terms
Linear Algebra Exam 1
Two linear systems of equations are eq…
The three elementary row operations are
We say that vector v is a linear combi…
A function f: X->Y is onto if
they have the same solution set.
Replace one row by the sum of itself and a multiple of anothe…
there exist scalars x1, x2,..., xn satisfying v = x1v1+x2v2+.…
for each y ∈ Y there is at least one x ∈ X with f(x) = y
Two linear systems of equations are eq…
they have the same solution set.
The three elementary row operations are
Replace one row by the sum of itself and a multiple of anothe…
21 terms
Linear algebra exam 1
8 Vector space properties
Vector space definition
Section 1.2 thms (8 properties related)
Subspace definition
1. Commutativity: u + v = v + u for all u, v ∈ V ;... 2. Associa…
V is a vs iff:... ∃! x+y ∈ V for every x,y ∈ V... ∃! cv ∈ V for eve…
1. x+y = y+x... => x=z... 2. i) 0 vector is unique... ii) inverse vec…
x+y ∈W... cx∈W... 0vector∈W OR W is nonempty
8 Vector space properties
1. Commutativity: u + v = v + u for all u, v ∈ V ;... 2. Associa…
Vector space definition
V is a vs iff:... ∃! x+y ∈ V for every x,y ∈ V... ∃! cv ∈ V for eve…
11 terms
Linear Algebra Test 1
consistent
inconsistent
row equivalent
Echelon form
when the system of equations has at least one solution
when the system of equations has no solutions
the systems have exactly the same solution set; Two matrices…
pivots can be any number; every all zero row of M is below ev…
consistent
when the system of equations has at least one solution
inconsistent
when the system of equations has no solutions
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