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Study sets matching "linear algebra chapter 1"

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
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.
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
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…
16 terms
Linear Algebra Chapter 1.1
Three elementary row operations
Inconsistent
consistent
Every elementary row operation is reve…
1) replace one row by the sum of itself and a multiple of ano…
no solution
1) one solution only... 2) infinitely many solutions
True
Three elementary row operations
1) replace one row by the sum of itself and a multiple of ano…
Inconsistent
no solution
57 terms
Linear Algebra Chapter 1
Linear equation
Linear system
linear system solution
Equivalent linear systems
an equation that can be written in the form a_1 x_1+a_2 X_2+.…
a collection of one or more linear equations involving the sa…
a list (s_1, s_2,...,s_n) of numbers that makes each equation…
Linear systems are equivalent if they have the same solution…
Linear equation
an equation that can be written in the form a_1 x_1+a_2 X_2+.…
Linear system
a collection of one or more linear equations involving the sa…
19 terms
Linear Algebra Chapter 1
A matrix is in reduced form if:
If a system has more variables than eq…
3 Facts about Linear Equations
Addition and Scalar Multiplication
1: Any rows consisting entirely of zeros are grouped together…
Must have an infinite number of solutions
Given an linear equation we must have 1) no solution... 2) one s…
Let u = (u1.....un) and v = (v1.....vn) be elements of Rn and…
A matrix is in reduced form if:
1: Any rows consisting entirely of zeros are grouped together…
If a system has more variables than eq…
Must have an infinite number of solutions
68 terms
Linear Algebra Chapter 1
Linear Equation
System of Linear Equations
Solution
Solution Set
describes a firsrt order line in Rn equations of the form a1x…
collection of 1+ linear equations to be solved simultaneously
a set of real numbers s1...sn such that each equation holds i…
a set of all solutions in a system
Linear Equation
describes a firsrt order line in Rn equations of the form a1x…
System of Linear Equations
collection of 1+ linear equations to be solved simultaneously
46 terms
Linear Algebra Chapter 1
equivalent
solution set
consistent
inconsistent
Two linear systems are called equivalent if they have the sam…
the set of all possible solutions of a linear system
A system of linear equations is said to be consistent if it h…
A system of linear equations is inconsistent if it has no sol…
equivalent
Two linear systems are called equivalent if they have the sam…
solution set
the set of all possible solutions of a linear system
22 terms
Linear Algebra Chapter 1
A linear system is consistent iff...
If a linear system is consistent then…
Linear Combination
Span(v1...vp)
The rightmost column of the augmented matrix is not a pivot c…
A unique solution w/ no free variables or infinitely many sol…
Given vectors *v*1,*v*2,...vp ven vectors *v*1,*v*2,...vp in R^n and given scalars c1,c…
Is the collection of all vectors that can be written in the f…
A linear system is consistent iff...
The rightmost column of the augmented matrix is not a pivot c…
If a linear system is consistent then…
A unique solution w/ no free variables or infinitely many sol…
10 terms
Linear Algebra Chapter 1
Consistent
Inconsistent
m x n Matrix
Trivial Solution
One or infinitely many solutions
No solution
m rows and n columns
The zero solution
Consistent
One or infinitely many solutions
Inconsistent
No solution
16 terms
Chapter 1 - Linear Algebra
Definition of equality
Definition of a subspace
Definition of a transpose
Definition of a symmetric matrix
Two m ⨉ n matrices are equal if all of their corresponding en…
A subset W of a vector space V over a field F is called a sub…
The transpose of At of an m ⨉ n matrix A is the n ⨉ m matrix…
A symmetric matrix is a matrix A such that At = A.
Definition of equality
Two m ⨉ n matrices are equal if all of their corresponding en…
Definition of a subspace
A subset W of a vector space V over a field F is called a sub…
33 terms
Linear Algebra Chapter 1
Linear Equation
System of Linear Equations
Solutions
Consistent
a1x1+a2x2+...+anxn=b
a11x1+a12x2+...+a1nxn=b1... a21x1+a22x2+...+a2nxn=b2... . ... .... .…
a sequence of n numbers that satisfies the equation.
A linear equation with a solution.
Linear Equation
a1x1+a2x2+...+anxn=b
System of Linear Equations
a11x1+a12x2+...+a1nxn=b1... a21x1+a22x2+...+a2nxn=b2... . ... .... .…
26 terms
Linear Algebra Chapter 1.7
linearly independent
linearly dependent
if there is a free variable...
theorem 7
{v1, v2, v3,... vn} is linear independent if x1v1 + x2v2 + x3…
{v1, v2, v3,... vn} is linear dependent if x1v1 + x2v2 + x3v3…
linearly dependent
if the set is linearly dependent at least one of the vectors…
linearly independent
{v1, v2, v3,... vn} is linear independent if x1v1 + x2v2 + x3…
linearly dependent
{v1, v2, v3,... vn} is linear dependent if x1v1 + x2v2 + x3v3…
14 terms
Linear Algebra Chapter 1.8
A linear transformation is a special t…
If A is an m x n matrix, then the rang…
If A is a 3x5 matric and T is a transf…
Every linear transformation is a matri…
True. The transformation is a function from one set of vector…
False. The range of T is the set of linear combinations of th…
False. The domain of T is Rn when A has n columns
False. Every matrix transformation is a linear transformation…
A linear transformation is a special t…
True. The transformation is a function from one set of vector…
If A is an m x n matrix, then the rang…
False. The range of T is the set of linear combinations of th…
14 terms
Linear Algebra Chapter 1.3
span
An example of a linear combination of…
The solution set of the linear system…
The set span{u,v} is always visualized…
the collection of all possible linear combinations of v1, v2,…
True
True
False, if the vectors are linearly dependent then their span…
span
the collection of all possible linear combinations of v1, v2,…
An example of a linear combination of…
True
20 terms
Linear Algebra Chapter 1
‖x‖
Direction of segment AB
AB = CD if
cx
‖x‖ = √(x₁²+x₂²+...)
= (b₁-a₁, b₂-a₂, ...)
(b₁-a₁, b₂-a₂, ...) = (d₁-c₁, d₂-c₂, ...)
= (cx₁, cx₂, ...)
‖x‖
‖x‖ = √(x₁²+x₂²+...)
Direction of segment AB
= (b₁-a₁, b₂-a₂, ...)
42 terms
Linear Algebra: Chapter 1
Every vector space contains a zero vec…
A vector space may have more than one…
In any vector space, ax=bx implies tha…
In any vector space, ax=ay implies tha…
True
False
False
False
Every vector space contains a zero vec…
True
A vector space may have more than one…
False
13 terms
Linear Algebra (Chapter 1)
In a system of equations, each equatio…
In a system of equations, each variabl…
what are the 3 elementary transformati…
systems of equations are solved by
restriction
degree of freedom
1. add rows together... 2. switch order of rows... 3. multiply each…
systematically eliminating variables
In a system of equations, each equatio…
restriction
In a system of equations, each variabl…
degree of freedom
23 terms
Linear Algebra Chapter 1
Elimination and Substitution
always
infinite
Linear equation
What are the two ways to solve for a linear system?
Homogeneous linear equations _________ have a solution
If two lines have the same slope, then the system has _____ a…
A ____ ____ of the variables x1,...,xn has the form... a1x1 + a…
Elimination and Substitution
What are the two ways to solve for a linear system?
always
Homogeneous linear equations _________ have a solution
9 terms
Linear Algebra Chapter 1
how to tell if b can be written as a l…
span {v1...vp} is...
Matrix equation
A(u+v)=
combine into augmented matrix and see if the matrix is consis…
the collection of all vectors which can be written in the for…
[v1 v2 v3][x1 x2 x3] ... Ax=b
Au + Av
how to tell if b can be written as a l…
combine into augmented matrix and see if the matrix is consis…
span {v1...vp} is...
the collection of all vectors which can be written in the for…
79 terms
Linear Algebra Chapter 1
Linear combination
Linear equation
System of linear equations
Solution
A _____ of A1, A2, ... An is an expression of the form :... c1A1…
A _____ is a linear cobination of unknowns set equal to a con…
A finite set of linear equations that all use the same variab…
A _____ to a system of linear equations is an assignment of v…
Linear combination
A _____ of A1, A2, ... An is an expression of the form :... c1A1…
Linear equation
A _____ is a linear cobination of unknowns set equal to a con…
10 terms
Linear Algebra Chapter 1
THEOREM 7: The inverse of a matrix, if…
THEOREM 9: Let A and B be n × n invert…
THEOREM 10 If the n × n matrix A is in…
THEOREM 11 If A is an m × n matrix, th…
Proof: Assume that the square matrix A has an inverse and tha…
Proof Using the properties of matrix multiplication, we have…
In certain cases we can find the solution of a linear system…
If a homogeneous linear system Ax = 0 is such that A is inver…
THEOREM 7: The inverse of a matrix, if…
Proof: Assume that the square matrix A has an inverse and tha…
THEOREM 9: Let A and B be n × n invert…
Proof Using the properties of matrix multiplication, we have…
26 terms
Linear Algebra Chapter 1
Theorem 1: Uniqueness of the Reduced E…
Theorem 2: Existence and Uniqueness Th…
Theorem 3: The Matrix Equation
Theorem 4: Existence of Solutions
Each matrix is row equivalent to one and only one reduced ech…
A linear system is consistent if and only if the rightmost co…
If A is an m x n matrix, with columns a1...an, and if b is in…
Let A be a m x n matrix. Then the following statements are lo…
Theorem 1: Uniqueness of the Reduced E…
Each matrix is row equivalent to one and only one reduced ech…
Theorem 2: Existence and Uniqueness Th…
A linear system is consistent if and only if the rightmost co…
48 terms
Linear Algebra Chapter 1
linear equation
linear system
solution
types of solutions
variables x1, x2, x3... variables have to stand alone
collection of linear equations with same variables
list of numbers that make each equation true when substituted…
no solution (lines parallel), one solution (lines cross) or i…
linear equation
variables x1, x2, x3... variables have to stand alone
linear system
collection of linear equations with same variables
41 terms
Linear Algebra Chapter 1
Theorem 1: Uniqueness of the Reduced E…
Theorem 2: Existence and Uniqueness Th…
Using Row Reduction to Solve a Linear…
Zero Vector
Each matrix is row equivalent to one and only one reduced ech…
A linear system is consistent if and only if the rightmost co…
Write the augmented matrix of the system
Vectors whose entries are zero
Theorem 1: Uniqueness of the Reduced E…
Each matrix is row equivalent to one and only one reduced ech…
Theorem 2: Existence and Uniqueness Th…
A linear system is consistent if and only if the rightmost co…
12 terms
Linear Algebra Chapter 1
Row Echelon Form
Reduced Row Echelon Form
How many solutions?
leading variables
1. LC in every row is 1.... 2. LC in every row is to the right o…
SAME as REF except ... 4. All coefficients above LC's are zeroes.
- if zero= nonzero---> no solution... - # of LC= # of variables/…
variables that are represented by a column with a leading one
Row Echelon Form
1. LC in every row is 1.... 2. LC in every row is to the right o…
Reduced Row Echelon Form
SAME as REF except ... 4. All coefficients above LC's are zeroes.
29 terms
Linear Algebra Chapter 1
linear equation
system of linear equations
solution
consistent system of linear equations
an equation that can be written in the form a₁x₁+a₂x₂+...
a collection of one or more linear equations using the same v…
a list of numbers (s₁,s₂...) that makes each equation a true…
a system of linear equations that has either one or infinitel…
linear equation
an equation that can be written in the form a₁x₁+a₂x₂+...
system of linear equations
a collection of one or more linear equations using the same v…
16 terms
Linear Algebra Chapter 1
N; Z; Q; R
x ∈ X
X ⊆ Y
X ⊂ Y
the sets of natural numbers, integers, rational numbers,... and…
denotes that x is a member of the set X
X is a subset of Y
X is a proper subset of Y
N; Z; Q; R
the sets of natural numbers, integers, rational numbers,... and…
x ∈ X
denotes that x is a member of the set X
14 terms
Linear Algebra Chapter 1
m x n matrix
Is the system consistent?
Echelon Form
REF
m rows and n columns
Does at least one solution exist, if yes is it the only one,…
1. All nonzero rows are above any rows of all zeros... 2. Each l…
1. The leading entry in each nonzero row is 1... 2. Each leading…
m x n matrix
m rows and n columns
Is the system consistent?
Does at least one solution exist, if yes is it the only one,…
18 terms
Linear Algebra Chapter 1.5
Non-Trivial Solution
Homogeneous equation
Describe x = p + tv
Trivial Solution
The homogenous equation Ax=0 has a nontrivial solution if and…
An equation in the form Ax=0, possibly written as a vector eq…
a line through p that is parallel to v
For a system of the form Ax=0 a zero solution
Non-Trivial Solution
The homogenous equation Ax=0 has a nontrivial solution if and…
Homogeneous equation
An equation in the form Ax=0, possibly written as a vector eq…
12 terms
Linear Algebra Chapter 1.4
The equation Ax = b is referred to as…
A vector b is a linear combination of…
The equation Ax = b is consistent if t…
The first entry in the product Ax is t…
False. This equation is the general form of a matrix equation
True. This equation has a solution if and only if b is a line…
Ax = b is consistent if the coefficient matrix [A b] has a pi…
True
The equation Ax = b is referred to as…
False. This equation is the general form of a matrix equation
A vector b is a linear combination of…
True. This equation has a solution if and only if b is a line…
9 terms
Linear Algebra Chapter 1.2
In some cases, a matrix may be row red…
The row reduction algorithm applies to…
A basic variable in a linear system is…
Finding a parametric description of th…
False. each matrix is row equivalent to one and only one redu…
False. The row reduction algorithm applies to any matrix whet…
True. The variables corresponding to pivot columns in an augm…
True.
In some cases, a matrix may be row red…
False. each matrix is row equivalent to one and only one redu…
The row reduction algorithm applies to…
False. The row reduction algorithm applies to any matrix whet…
29 terms
Linear Algebra Chapter 1
linear equation
system of linear equations
solution to linear system
solution set
equation that can be written in the form: a1x1 + a2x2+... anx…
a collection of linear equations involving x's that you don't…
list (s1, s2, s3, ... sn) which satisfies each equation when…
set of all possible solutions
linear equation
equation that can be written in the form: a1x1 + a2x2+... anx…
system of linear equations
a collection of linear equations involving x's that you don't…
15 terms
Linear Algebra - Chapter 1.1
a __ is a collection of one or more li…
a __ for a SLE is a list of numbers su…
two systems are called __ if they have…
three of the elementary row operations
system of linear equations
solution
equivalent
replacement interchange scaling
a __ is a collection of one or more li…
system of linear equations
a __ for a SLE is a list of numbers su…
solution
Linear Algebra Chapter 1.9
If T: Rn maps to Rm is a linear transf…
A linear transformation T: Rn maps ont…
If T: R2 to R2 rotates vectors about t…
When two linear transformations are pe…
Then there exists a unique matrix A such that T(x) = Ax for a…
True. T is completely determined by what it does to the colum…
True
False. Linear Transformations preserve the operations of vect…
If T: Rn maps to Rm is a linear transf…
Then there exists a unique matrix A such that T(x) = Ax for a…
A linear transformation T: Rn maps ont…
True. T is completely determined by what it does to the colum…
15 terms
Linear Algebra - Chapter 1.2
the __ of a row refers to the leftmost…
a __ in a matrix A is a location in A…
a __ is a column of A that contains a…
pivot positions represent __
leading entry
pivot position
pivot column
basic variables
the __ of a row refers to the leftmost…
leading entry
a __ in a matrix A is a location in A…
pivot position
16 terms
Linear Algebra - Chapter 1.3
a matrix with only one column is calle…
two vectors in R^2 are __ if and only…
the vector whose entries are all zero…
given vectors v_1, v_2, ... , v_p in R…
column vector
equivalent
zero vector
linear combination weights
a matrix with only one column is calle…
column vector
two vectors in R^2 are __ if and only…
equivalent
10 terms
Linear Algebra Chapter 1.1
C.The solution set is empty.
C. The solution set contains one​ solu…
A. The matrix is the augmented matrix…
Is the statement​ "Two matrices are ro…
A. The solution set has exactly one​ element, left parenthesi…
A. The solution has infinitely many elements.... B. The solutio…
Determine the​ value(s) of h such that the matrix is the augm…
C. False, because if two matrices are row equivalent it means…
C.The solution set is empty.
A. The solution set has exactly one​ element, left parenthesi…
C. The solution set contains one​ solu…
A. The solution has infinitely many elements.... B. The solutio…
27 terms
Chapter 1: Linear Equations in Linear Algebra
Linear Equation
Linear System
Solution
Solution Set
An equation that can be written in the form:... a1x1+a2x2+...+an…
A collection of one or more linear equations involving the sa…
A list (s1, s2, ..., sn) of numbers that makes each equation…
The set of all possible solutions for a given system
Linear Equation
An equation that can be written in the form:... a1x1+a2x2+...+an…
Linear System
A collection of one or more linear equations involving the sa…
54 terms
Chapter 1: Linear Equations in Linear Algebra
When is a system of linear equations c…
When is a system of linear equations i…
When do 2 linear systems have the same…
(Row) Echelon Form
When it has either one solution or infinitely many solutions
When it has no solution
If their augmented matrices are row equivalent
1) All nonzero rows are above any rows of all zeros... 2) Each l…
When is a system of linear equations c…
When it has either one solution or infinitely many solutions
When is a system of linear equations i…
When it has no solution
30 terms
Linear Algebra Chapter 1 Terms
Consistent
Inconsistent
unique solution
row equivalent
a system of linear equations that has either 1 or infinte sol…
a system of linear equations that has no solution
the solutions when there is only one possible solutions for a…
when you can turn one matix into the other
Consistent
a system of linear equations that has either 1 or infinte sol…
Inconsistent
a system of linear equations that has no solution
10 terms
Algebra 1 Chapter 1 Solving Linear Equations
conjecture
theorem or rule
equation
linear equation in one variable
an unproven statement about a general mathematical concept
a proven statement about a general mathematical concept
a statement that two expressions are equal
an equation that can be written in the form ax + b = 0, where…
conjecture
an unproven statement about a general mathematical concept
theorem or rule
a proven statement about a general mathematical concept
8 terms
Khan Academy. Linear Algebra. Chapter 1.
Vector
Physics definition
Computer Science definition
Mathematics definition
Is abstraction, the meaning of it depends on the particular a…
In this field of study, where vector represents something tha…
In this field of study, where vector represents a pair (or ev…
In this field of study, where vector is abstracted, so it rep…
Vector
Is abstraction, the meaning of it depends on the particular a…
Physics definition
In this field of study, where vector represents something tha…
Linear Algebra Chapter 1 Vocab
linear equation
linear system
solution to a linear system with n var…
solution set
an equation of the variables x1, x2, . . . , xn that can be w…
a collection of one or more linear equations using the same v…
a list of n numbers (s1, s2, . . . sn)
the set of all possible solutions to a given linear system
linear equation
an equation of the variables x1, x2, . . . , xn that can be w…
linear system
a collection of one or more linear equations using the same v…
8 terms
Chapter 1 definitions Advanced linear algebra
vector space
subspace
span
spans (generates)
a space V over a field F consists of a set which two operatio…
A subset W of a vector space V over a field F is a subspace o…
Let S be a nonempty subset of a vector space V. the span of S…
A subset S of a vector space V generates (or spans) V if span…
vector space
a space V over a field F consists of a set which two operatio…
subspace
A subset W of a vector space V over a field F is a subspace o…
12 terms
Chapter 1 theorems Advanced Linear algebra
Theorem 1.1 (Cancellation Law for Vect…
Theorem 1.2
Theorem 1.3
Theorem 1.4
If x,y, and z are vectors in a vector space V such that x+z=y…
In any vector space V, the following statements are true:... a)…
Let V be a vector space and W a subset of V. Then W is a subs…
Any intersection of subspaces of a vector space V is a subspa…
Theorem 1.1 (Cancellation Law for Vect…
If x,y, and z are vectors in a vector space V such that x+z=y…
Theorem 1.2
In any vector space V, the following statements are true:... a)…
13 terms
Linear Algebra Chapter 1 Terms
What is a matrix?
What is a square matrix?
What is a diagonal matrix?
What is Equality
A matrix is a rectangular array of numbers or... symbols.
A matrix with the same number of columns and rows, also known…
A square matrix, N x N, that at entries that are not on the d…
Two matrices are equal if they are the same size and all the…
What is a matrix?
A matrix is a rectangular array of numbers or... symbols.
What is a square matrix?
A matrix with the same number of columns and rows, also known…
12 terms
Linear Algebra: Chapter 1 (Vector spaces, linear maps and rings)
Definition of a Field and its Characte…
Properties of a vector space over a fi…
Definition of 'Linear Independence'
Definition of 'Spanning'
...
...
...
...
Definition of a Field and its Characte…
...
Properties of a vector space over a fi…
...
31 terms
Algebra 1 Linear Review
D
A
D
B
D
A
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