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Study sets matching "linear algebra"

126 terms
Linear Algebra
Reduced row echelon form conditions
Theorem "dimension characterizes isomo…
Define "W is isomorphic to V"
Theorem stating "isomorphic to" is an…
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 iso…
There exists an isomorphism T:W->V
"isomorphic to" is an equivalence relation for subspaces V wi…
Reduced row echelon form conditions
1. leading entry (if there is one) is 1 2. if column contains…
Theorem "dimension characterizes isomo…
For subspaces V within R^n and W within R^m, we have V is iso…
66 terms
Linear Algebra
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 en, [1, 0, 0, 0...0] to the nth number. So in R3…
Two vectors, when added together, are still in the same subsp…
When a vector is multiplied by a scalar, it's still in the sa…
Scalar multiplication
Multiplication of vectors by t, such that it becomes t(v1), t…
Standard Basis in Rn
Denoted by en, [1, 0, 0, 0...0] to the nth number. So in R3…
15 terms
Linear Algebra
Slope Intercept Form
System of linear equations
Point Slope Form
The answer to a system of linear equat…
y= mx + b
2x + y = 5 ... 3x - y = 6
y - y^1 = m(x - x^1)
Ordered Pair
Slope Intercept Form
y= mx + b
System of linear equations
2x + y = 5 ... 3x - y = 6
126 terms
Linear Algebra
Reduced row echelon form conditions
Theorem "dimension characterizes isomo…
Define "W is isomorphic to V"
Theorem stating "isomorphic to" is an…
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 iso…
There exists an isomorphism T:W->V
"isomorphic to" is an equivalence relation for subspaces V wi…
Reduced row echelon form conditions
1. leading entry (if there is one) is 1 2. if column contains…
Theorem "dimension characterizes isomo…
For subspaces V within R^n and W within R^m, we have V is iso…
66 terms
Linear Algebra
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 en, [1, 0, 0, 0...0] to the nth number. So in R3…
Two vectors, when added together, are still in the same subsp…
When a vector is multiplied by a scalar, it's still in the sa…
Scalar multiplication
Multiplication of vectors by t, such that it becomes t(v1), t…
Standard Basis in Rn
Denoted by en, [1, 0, 0, 0...0] to the nth number. So in R3…
67 terms
Linear algebra
What is a subspace and what are the re…
Column Space?
What is closure?
What is true for the span of a set of…
A subset (H) of a larger set of vectors(V) that has 3 propert…
denoted col(A)... the span of the columns of A.
When an operation such as addition is done with two numbers f…
For a set of vectors {v₁, v₂, v₃....} in Rⁿ... span{v₁,v₂,v₃...…
What is a subspace and what are the re…
A subset (H) of a larger set of vectors(V) that has 3 propert…
Column Space?
denoted col(A)... the span of the columns of A.
67 terms
Linear algebra
What is a subspace and what are the re…
Column Space?
What is closure?
What is true for the span of a set of…
A subset (H) of a larger set of vectors(V) that has 3 propert…
denoted col(A)... the span of the columns of A.
When an operation such as addition is done with two numbers f…
For a set of vectors {v₁, v₂, v₃....} in Rⁿ... span{v₁,v₂,v₃...…
What is a subspace and what are the re…
A subset (H) of a larger set of vectors(V) that has 3 propert…
Column Space?
denoted col(A)... the span of the columns of A.
67 terms
Linear algebra
What is a subspace and what are the re…
Column Space?
What is closure?
What is true for the span of a set of…
A subset (H) of a larger set of vectors(V) that has 3 propert…
denoted col(A)... the span of the columns of A.
When an operation such as addition is done with two numbers f…
For a set of vectors {v₁, v₂, v₃....} in Rⁿ... span{v₁,v₂,v₃...…
What is a subspace and what are the re…
A subset (H) of a larger set of vectors(V) that has 3 propert…
Column Space?
denoted col(A)... the span of the columns of A.
34 terms
Linear Algebra
linear equation
consistent system
inconsistent system
leading entry
An equation that can be written as a1x1 + a2x2 + ... = b; a1,…
Has one or infinitely many solutions
Has no solution
Leftmost non-zero entry in a non-zero row
linear equation
An equation that can be written as a1x1 + a2x2 + ... = b; a1,…
consistent system
Has one or infinitely many solutions
31 terms
Algebra 1 Linear Review
D
A
D
B
D
A
31 terms
Algebra 1 Linear Review
D
A
D
B
D
A
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
9 terms
Algebra Linear functions
Y- intercept
Slope
Slope- intercept Form
Point- slope form
the y-coordinate of a point where a graph crosses the y-axis
the steepness of a line on a graph, equal to its vertical cha…
an equation written in the form y=mx+b is in slope-intercept…
y-y1 = m(x-x1), where m is the slope and (x1,y1) is the point…
Y- intercept
the y-coordinate of a point where a graph crosses the y-axis
Slope
the steepness of a line on a graph, equal to its vertical cha…
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.…
92 terms
linear algebra
matrix notation for an entry
augmented matrix
row-echelon form
Gaussian elimination
(A)ij means the entry for A in the i row and the j column. Or…
where you have the solutions set up as another column; the ri…
Any rows entirely of 0 are at the bottom. Each row's first no…
*I think* where you do row operations starting at the top and…
matrix notation for an entry
(A)ij means the entry for A in the i row and the j column. Or…
augmented matrix
where you have the solutions set up as another column; the ri…
66 terms
Linear Algebra
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 en, [1, 0, 0, 0...0] to the nth number. So in R3…
Two vectors, when added together, are still in the same subsp…
When a vector is multiplied by a scalar, it's still in the sa…
Scalar multiplication
Multiplication of vectors by t, such that it becomes t(v1), t…
Standard Basis in Rn
Denoted by en, [1, 0, 0, 0...0] to the nth number. So in R3…
60 terms
Linear Algebra
Solving if something is in a Span
Span Theorems
Span
Subset
By definition of span, we want to know if there are scalars x…
Theorem 1: The subspace spanned by a non-empty subset S of a…
The set of all linear combinations of the vectors.... or ... The…
A subset is a set of vectors. Assume a subset V∈ℜn... Can be a…
Solving if something is in a Span
By definition of span, we want to know if there are scalars x…
Span Theorems
Theorem 1: The subspace spanned by a non-empty subset S of a…
100 terms
Linear Algebra
linear equation
system of linear equations
solution set
equivalent linear systems
an equation that can be written in the form a1x1 + a2x2 + ...…
a collection of one or more linear equations involving the sa…
a list of numbers that makes each equation in a system a true…
linear systems with the same solution set
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 involving the sa…
42 terms
Linear Algebra
Elementary Row Operations
Row Equivalency
Two equivalent equivalent augmented ma…
Two Fundamental Questions
1. Replacement... 2. Scaling... 3. Interchange
A is row equivalent to B if we can get from A to B via a sequ…
solution set
1. Is the system consistent... 2. If the solution exists, is the…
Elementary Row Operations
1. Replacement... 2. Scaling... 3. Interchange
Row Equivalency
A is row equivalent to B if we can get from A to B via a sequ…
97 terms
Linear algebra
Definition of a vector space
8 axioms of a vector space
Notation for a vector space with n dim…
Definition of a linear combination of…
V (set of elements: x, y, u, v) is called a vector space of K…
1) *u, v, w* ∈ V, (*u* + *v*) + *w* = *u* + (*v* + *w*) (Asso…
ℝⁿ
*x* is a linear combination of *v₁, ... vₙ* if for *v₁, ... v…
Definition of a vector space
V (set of elements: x, y, u, v) is called a vector space of K…
8 axioms of a vector space
1) *u, v, w* ∈ V, (*u* + *v*) + *w* = *u* + (*v* + *w*) (Asso…
34 terms
linear algebra
3.1 Two equivalent vectors must have t…
3.1 The vectors (a,b) and (a,b,0) are…
3.1 If k is a scalar and v is a vector…
3.1 The vectors v+(u+w) and (w+v)+u ar…
F (옮겨서 시점이 달라져도 같은 벡터로 취급한다.)
F (일단 기본적으로 차원이 같아야 함.)
F (벡터가 평행하다는 것은 굳이 방향이 같을 필요는 없음. 방향은 반대더라도 됨.)
T
3.1 Two equivalent vectors must have t…
F (옮겨서 시점이 달라져도 같은 벡터로 취급한다.)
3.1 The vectors (a,b) and (a,b,0) are…
F (일단 기본적으로 차원이 같아야 함.)
42 terms
Linear Algebra
Solution of an SLE
Consistent SLE
Inconsistent SLE
Equivalent linear system
A SLE has (1) no solution, or (2) exactly one solution, or (3…
SLE has one or infinitely many solutions.
SLE has no solution.
A SLE that have the same solution set
Solution of an SLE
A SLE has (1) no solution, or (2) exactly one solution, or (3…
Consistent SLE
SLE has one or infinitely many solutions.
15 terms
Linear Algebra
Elementary Matrix
property of Elementary Matrix
Cramer's Rule2
subspace of Rn
what an n x n matrix E is called if we can obtain E from I, b…
each of these elementary row operations can be implemented by…
(a) the sum of any pair of vectors in the set lies in the set…
Elementary Matrix
what an n x n matrix E is called if we can obtain E from I, b…
property of Elementary Matrix
each of these elementary row operations can be implemented by…
16 terms
Linear algebra
Matix
Elements of the matrix
Submatrix
Square matrix
A rectangular array of numbers.
The numbers in the array
A matrix composed of a array of numbers taken from a larger m…
When there are equal numbers of rows as Columns
Matix
A rectangular array of numbers.
Elements of the matrix
The numbers in the array
51 terms
Linear Algebra
Linear combination
Linearly independent
Theorem 5.1.1
elementary vectors
an expression of the form c1v1+c2v2+...+cnvn where all the ci…
a set of vectors in LI if the only way to write the zero vect…
A set of vectors is LD iff one of the vectors can be written…
vectors that have exactly one component equal to 1 and all ot…
Linear combination
an expression of the form c1v1+c2v2+...+cnvn where all the ci…
Linearly independent
a set of vectors in LI if the only way to write the zero vect…
19 terms
Linear Algebra
Col A
Nul A
Rank
Col A(T)
Pivot Columns
Free variables in parametric form
number of pivot columns
Pivot Row
Col A
Pivot Columns
Nul A
Free variables in parametric form
80 terms
Linear Algebra
Linear Equation
System of Linear Equations
Solution
Solution Set
A LINEAR EQUATION in variables x₁,x₂,....,xⁿ (for n a positiv…
A SYSTEM OF LINEAR EQUATIONS is a collection of one or more l…
A SOLUTION to a linear system is a list (s₁,s₂,...,sⁿ) of num…
The set of all solutions to a linear system is called the SOL…
Linear Equation
A LINEAR EQUATION in variables x₁,x₂,....,xⁿ (for n a positiv…
System of Linear Equations
A SYSTEM OF LINEAR EQUATIONS is a collection of one or more l…
10 terms
Linear Algebra
the image of at least one x ∈ R.
T(u + v) = T(u) + T(v)
T(cu) = cT(u)
row equivalent
A transformation T : Rn → Rm is one-to-one if each b ∈ Rm is…
_________ for all u, v in the domain of T.
_________ for all scalars c and all vectors u in the domain o…
two matrices are _________ if one can be changed to the other…
the image of at least one x ∈ R.
A transformation T : Rn → Rm is one-to-one if each b ∈ Rm is…
T(u + v) = T(u) + T(v)
_________ for all u, v in the domain of T.
28 terms
Linear Algebra: Vector Spaces
What is a subspace and what are the re…
Column Space?
What is closure?
What is true for the span of a set of…
A subset (H) of a larger set of vectors(V) that has 3 propert…
denoted col(A)... the span of the columns of A.
When an operation such as addition is done with two numbers f…
For a set of vectors {v₁, v₂, v₃....} in Rⁿ... span{v₁,v₂,v₃...…
What is a subspace and what are the re…
A subset (H) of a larger set of vectors(V) that has 3 propert…
Column Space?
denoted col(A)... the span of the columns of A.
128 terms
Linear algebra
What is a subspace and what are the re…
Column Space?
column rank
Row space
A subset (H) of a larger set of vectors(V) that has 3 propert…
denoted col(A)... the span of the columns of A.... the basis are th…
cr:=dim(column space)
denoted row(A)... the span of rows of A... the basis is all nonzero…
What is a subspace and what are the re…
A subset (H) of a larger set of vectors(V) that has 3 propert…
Column Space?
denoted col(A)... the span of the columns of A.... the basis are th…
Linear Algebra
Suppose we have a final solution set o…
How are two linear systems equivalent?
How do you solve a linear system?
A matrix with size 3x4 is: ___________…
infinite
They share the same solution sets
Replace the l.s. with one that is equivalent but "simpler", t…
augumented matrix of *
Suppose we have a final solution set o…
infinite
How are two linear systems equivalent?
They share the same solution sets
108 terms
Friedell Linear Algebra Vocabulary
Order of Operations
Inequality
Like Terms
Equivalent expressions
Ch. 1 PEMDAS the order in which you solve an equation. (Paren…
Ch. 1 An open sentence that contains the symbols < >
Ch. 1 Terms that contain the same variables, with correspondi…
Ch. 1 Expressions that simplify to the same number.
Order of Operations
Ch. 1 PEMDAS the order in which you solve an equation. (Paren…
Inequality
Ch. 1 An open sentence that contains the symbols < >
13 terms
Linear Algebra
A finite basis for V of Field F is a f…
Let S be a nonempty subset of a vector…
Let V & W be vector spaces over field…
A nonempty finite subset {u1, u2,.....…
Finite Basis
Span
Linear Transformations (Operators)
Linear Dependent
A finite basis for V of Field F is a f…
Finite Basis
Let S be a nonempty subset of a vector…
Span
6 terms
Linear Algebra
Truculent
Rapacious
Misanthrope
Craven
Eager or quick to argue or fight; aggressively defiant. ... Syn:…
Person who is abnormally anxious about their health.... Syn: Val…
General hatred, distrust and contempt of the human species or…
A cowardly person; contemptibly lacking in courage. "A craven…
Truculent
Eager or quick to argue or fight; aggressively defiant. ... Syn:…
Rapacious
Person who is abnormally anxious about their health.... Syn: Val…
24 terms
Linear Algebra
What is the null space of matrix A? Wh…
A subspace of Rⁿ is any set H in Rⁿ th…
What is the column space of matrix A?
The set {e₁; ...; eη} is called the __…
The null space of a matrix A is the set Nul A of all solution…
a. The zero vector is in H.... b. For each u and v in H, the sum…
The column space of a matrix A is the set Col A of all linear…
standard basis
What is the null space of matrix A? Wh…
The null space of a matrix A is the set Nul A of all solution…
A subspace of Rⁿ is any set H in Rⁿ th…
a. The zero vector is in H.... b. For each u and v in H, the sum…
14 terms
Linear Algebra
What is a vector?
What is a matrix?
What is the vector space?
What are the 8 axioms of vector space?
Vector: a quantity with more than one element (more than one…
Put simply, a rectangular array of numbers, symbols, expressi…
A vector space (also called a linear space) is a collection o…
What is a vector?
Vector: a quantity with more than one element (more than one…
What is a matrix?
Put simply, a rectangular array of numbers, symbols, expressi…
30 terms
Linear Algebra: Matrices
Matrix Dimensions are given by? What d…
Scalar multiplication of a matrix?
How are Matrices added/subtracted?
What is the transpose of a matrix and…
rows x columns (m x n)... The rows are the number of equations…
A number that multiplies each number in the matrix.
Just add/subtract each corresponding entry.... They must theref…
You just swap rows with columns.
Matrix Dimensions are given by? What d…
rows x columns (m x n)... The rows are the number of equations…
Scalar multiplication of a matrix?
A number that multiplies each number in the matrix.
22 terms
Linear algebra expressions
nothing, it is in standard form
9, 7, 4
10
3
What must to be done to put this expression in standard form?
What are the coefficients?
What is the constant?
What is the degree?
nothing, it is in standard form
What must to be done to put this expression in standard form?
9, 7, 4
What are the coefficients?
47 terms
Friedell Linear Algebra Vocabulary
Order of Operations
Inequality
Like Terms
Equivalent expressions
PEMDAS the order in which you solve an equation.
An open sentence that contains the symbols < >
Terms that contain the same variables, with corresponding var…
Expressions that denote the same number.
Order of Operations
PEMDAS the order in which you solve an equation.
Inequality
An open sentence that contains the symbols < >
31 terms
Algebra 1 Linear Review
D
A
D
B
D
A
31 terms
Algebra 1 Linear Review
D
A
D
B
D
A
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…
10 terms
Linear Algebra
Coefficient matrix
Augmented Matrix
Span
Linearly independent
only includes matrix
includes answers
intersection of all subspace containing that set.
No free variables
Coefficient matrix
only includes matrix
Augmented Matrix
includes answers
83 terms
Linear algebra
Consistant
Coefficient matrix
Augmented matrix
Echelon matrix
A linear equation set with one or infinite solutions (imterse…
Only the coefficient of linear equations
Shows coefficients and adds a column with constants from righ…
1. All non zero rows are above all zero rows... 2. Each leading…
Consistant
A linear equation set with one or infinite solutions (imterse…
Coefficient matrix
Only the coefficient of linear equations
11 terms
Linear Algebra
domain
codomain
range
image
...
...
...
...
domain
...
codomain
...
12 terms
Linear Algebra
Linear Dependence
Linear Independence
Linear Combination
Span
When a vector in a set can be expressed as a linear combinati…
When none of the vectors in a set can be expressed as a linea…
c1v1 + c2v2 + c3v3 ... cnvn | ci is a member of R for i < n+1…
The set of all of the vectors that can be created by linear c…
Linear Dependence
When a vector in a set can be expressed as a linear combinati…
Linear Independence
When none of the vectors in a set can be expressed as a linea…
61 terms
Linear Algebra
consistent system
inconsistent system
row operations
row echelon form
a system with at least one solution
a system with no solutions
multiply an equation by a non zero constant; add a multiple o…
first non zero entry in each row is 1 and every "leading 1" h…
consistent system
a system with at least one solution
inconsistent system
a system with no solutions
30 terms
linear algebra
linear system
standard form
homogeneous
inhomogeneous
set of linear equations
variables on left... constants on right
all 0s on right-hand side
at least one non-zero constant on right-hand side
linear system
set of linear equations
standard form
variables on left... constants on right
12 terms
Linear Algebra
Square matrix
diagonal
identity
zero
n × n
is a square matrix where the only non-zero entries are the on…
is a diagonal matrix where the main diagonal entries are ones
matrix where all entries are zero
Square matrix
n × n
diagonal
is a square matrix where the only non-zero entries are the on…
99 terms
Linear Algebra
A linear equation in the variables x_1…
A solution of the system
The set of all possible solutions is c…
Two linear systems that have the same…
(a_1) X1 + (a_2) X2 + ... + (a_n) * Xn
is a list (s_1, s_2, ... , s_n) of numbers that makes each eq…
solution set
are called "equivalent"
A linear equation in the variables x_1…
(a_1) X1 + (a_2) X2 + ... + (a_n) * Xn
A solution of the system
is a list (s_1, s_2, ... , s_n) of numbers that makes each eq…
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