Study sets matching "linear math algebra"

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Study sets matching "linear math algebra"

GRE MATH Linear Algebra
invertible matrix
If A and B are invertible matrices of…
formula meaning A^n is invertible
If A is an invertible matrix, then
all A such that there exists A^-1 such that AA^-1 = A^-1A = I
AB is invertible, and (AB)^-1 = B^-1A^-1
(A^n)^-1 = (A^-1)^n
A^-1 is invertible, and A^n is invertible
invertible matrix
all A such that there exists A^-1 such that AA^-1 = A^-1A = I
If A and B are invertible matrices of…
AB is invertible, and (AB)^-1 = B^-1A^-1
101 terms
Math vocabulary for linear algebra
Variables... Section 1-1 variables and ex…
Algebraic expressions... Section 1-1 vari…
Factors... Section 1-1 variables and expr…
Product... Section 1-1 variables and expr…
Are symbols used to represent unspecific numbers or values. A…
Consists of one or more numbers and variables along with one…
The quantities being multiplied... Example 2x4
The outcome of what is being multiplied... Example 2x4 product i…
Variables... Section 1-1 variables and ex…
Are symbols used to represent unspecific numbers or values. A…
Algebraic expressions... Section 1-1 vari…
Consists of one or more numbers and variables along with one…
Linear Algebra Math135
Consistent System
Inconsistent System
Consistent Dependent System
Free variable
The matrix system has a solution
Matrix has no solution (last row 0 0 | non zero )
Infinite systems (last row 0 0 | 0 )
x1 x2 x3 etc can be any number, no set equation, no pivot in…
Consistent System
The matrix system has a solution
Inconsistent System
Matrix has no solution (last row 0 0 | non zero )
MATH 3083: Linear Algebra
equivalent system of equations
A Homogeneous system
invertible matrix
singularity
when two system of equations have the same solution set
when the constants on the right hand side of the matrix are a…
when there exists a matrix B such that AB=BA=I.
when a nxn matrix does not have a multiplicative inverse.
equivalent system of equations
when two system of equations have the same solution set
A Homogeneous system
when the constants on the right hand side of the matrix are a…
28 terms
Math 31 (Linear Algebra)
Thm 1
Thm 2
Linear Independence
Linear Dependence
A linear system has at least one solution if & only if the ri…
Let A be an mxn matrix coefficient matrix. Then the following…
The columns of A are linearly independent if & only if the tr…
The columns of A are linearly dependent if there is a nontriv…
Thm 1
A linear system has at least one solution if & only if the ri…
Thm 2
Let A be an mxn matrix coefficient matrix. Then the following…
Linear Algebra MATH 313
Consistent System
Inconsistent System
Row Equivalent Matrices
Linear Combination
The system has either 1 solution or infinitely many
The system has no solutions
If a sequence of Elementary Row Operations can transform one…
Given vectors in R^n and scalars c1....cp, the vector b is gi…
Consistent System
The system has either 1 solution or infinitely many
Inconsistent System
The system has no solutions
30 terms
Math 110 - Linear Algebra
(VS1) ... Commutativity of Addition
(VS2)... Associativity of Addition
(VS3)... Zero Vector;... Identity Element of…
(VS4)... Inverse Elements of Addition
For all x, y in V, x+y = y+x
For all x, y, z in V (x+y)+z = x+ (y+z)
There exists an element in V denoted by 0 such that x+0=x for…
For each element x in V there exists and element y in V such…
(VS1) ... Commutativity of Addition
For all x, y in V, x+y = y+x
(VS2)... Associativity of Addition
For all x, y, z in V (x+y)+z = x+ (y+z)
23 terms
Linear Algebra (Math 4C)
System of Linear Equations
Solution of a System of Linear Equations
Solution Set of a System of Linear Equ…
Equivalent Systems
A collection of m equations with the same variable quantities…
The solution of a system is a list (s1, s2, s3, ..., sn) of n…
The set of all possible solutions for a linear system.
When two linear systems have the SAME solution set.
System of Linear Equations
A collection of m equations with the same variable quantities…
Solution of a System of Linear Equations
The solution of a system is a list (s1, s2, s3, ..., sn) of n…
32 terms
Math 1553 - linear algebra KNOW
x bar parallel is
two vectors are orthogonal to each oth…
magnitude is
In an LU decomposition, the columns of…
the orthogonal projection of x bar onto the line
u * v = 0
the square root of each component squared
the pivotal columns in original matrix
x bar parallel is
the orthogonal projection of x bar onto the line
two vectors are orthogonal to each oth…
u * v = 0
16 terms
Math 340 Linear Algebra
Linear system
What are two basic methods of solving…
Two systems of linear equations are eq…
How do we know the method of eliminati…
A set of linear equations.
Method of elimination and substitution.
If they have the same solutions.
Each step in the method of elimination gives an equivalent sy…
Linear system
A set of linear equations.
What are two basic methods of solving…
Method of elimination and substitution.
11 terms
Math 262 Linear Algebra
Scalar Projection
Vector Projection
‖vxw‖= ?
Scalar triple product
s = u·v/|v| = |u|cosθ
(v·d)/(‖d‖²) d
‖v‖‖w‖sinθ = Area of parallelogram determined by v and w.
u·(vxw) --> volume of parallelepiped
Scalar Projection
s = u·v/|v| = |u|cosθ
Vector Projection
(v·d)/(‖d‖²) d
6 terms
math2121 linear algebra
linear algebra
success
substitution
subtract something from
elimination
failure
multiplication
permutation
linear algebra
elimination
success
failure
Linear Algebra (Math 3A)
Linear equation
Linear system (system of linear equati…
Solutions
Solution set
Equation that can be written in the form a₁x₁+a₂x₂+... = b
Collection of one or more linear equations involving the same…
Numbers that make each equation true
Set of all possible solutions
Linear equation
Equation that can be written in the form a₁x₁+a₂x₂+... = b
Linear system (system of linear equati…
Collection of one or more linear equations involving the same…
Linear Algebra MATH 13
Elementary row operations
Equivalent systems of linear equations
Matrix in row-echelon form properties
Matrix in reduced row-echelon form pro…
1. Interchange two rows... 2. Multiply a row by a nonzero consta…
Systems that have the same solution set
1. Rows of only zeros are at the bottom... 2. For each row that'…
It's in row-echelon form and every column that has a leading…
Elementary row operations
1. Interchange two rows... 2. Multiply a row by a nonzero consta…
Equivalent systems of linear equations
Systems that have the same solution set
27 terms
Math 221 (Linear Algebra Terms)
Linear equation
Coefficients
System of linear equations or Linear s…
Solution of the system
In the variables x(1)...x(n) is an equation that can be writt…
Real or complex numbers
Collection of one or more linear equations involving the same…
A list (s(1),s(2),...,s(n)) of numbers that makes each equati…
Linear equation
In the variables x(1)...x(n) is an equation that can be writt…
Coefficients
Real or complex numbers
Linear Algebra Math 5 (lay)
system of linear equations
linear equation
linear system
solution of a system
a collection of one ore more linear equations involving the s…
equation of first order with integer coefficients
same as linear equation
a list of numbers that makes each equation of a linear system…
system of linear equations
a collection of one ore more linear equations involving the s…
linear equation
equation of first order with integer coefficients
Math 1553 Linear Algebra Final
if the augmented matrix (A|b) has a pi…
Ax = 0 is consistent
if m = n then Ta is onto
if m < n then Ax = b has infinitely ma…
False
True
Maybe
False
if the augmented matrix (A|b) has a pi…
False
Ax = 0 is consistent
True
8 terms
Linear Algebra Terms MATH 217
Vector Space
Subspace
Linear Transformation
Source & Target
A vector space is a set V , equipped with a rule for addition…
If V is a vector space, a subspace of V is a non-empty subset…
Let V and W be vector spaces. A linear transformation is a ma…
Suppose V −→ W is a linear transformation. The vector space V…
Vector Space
A vector space is a set V , equipped with a rule for addition…
Subspace
If V is a vector space, a subspace of V is a non-empty subset…
53 terms
Math 3333: Linear Algebra Final
If A is symmetric then the eigenvector…
All of the roots of the characteristic…
If A can be diagonalized then
If A can be diagonalized then there ex…
orthogonal
real numbers
the columns of P in P^(-1)AP = D are eigenvectors of A
a nonsingular matrix P such that P^(-1)AP is diagonal and the…
If A is symmetric then the eigenvector…
orthogonal
All of the roots of the characteristic…
real numbers
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
54 terms
MATH 307 Applied Linear Algebra
Hilbert-Schmidt norm (for a matrix) (a…
Matrix (operator) norm
Vandermonde matrix
Formula for the determinant of a Vande…
concatenate all the columns of the matrix end-to-end and take…
A measure of the maximum amount by which a matrix "stretches"…
Matrix used in Lagrangian interpolation (in 2D). Each row cor…
Plus or minus the product of the pairwise differences of ever…
Hilbert-Schmidt norm (for a matrix) (a…
concatenate all the columns of the matrix end-to-end and take…
Matrix (operator) norm
A measure of the maximum amount by which a matrix "stretches"…
38 terms
NST IA Maths: Linear Algebra
Vectors are linearly independent
Vector Space V is n-dimensional
Matrices can only be multiplied
Product cij
if the only solution of... a₁v₁ + ... + am vm = 0... is ai = 0, ∀i
dim V = n, if there exist linearly independent vectors, such…
if A is m x n and B is n x p
ⁿ∑ aik bkj
Vectors are linearly independent
if the only solution of... a₁v₁ + ... + am vm = 0... is ai = 0, ∀i
Vector Space V is n-dimensional
dim V = n, if there exist linearly independent vectors, such…
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…
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…
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…
5 terms
Dr. Math Explains Algebra Linear Equations
What are linear equations
Slope intercept form
Variables in a linear expression can o…
What is a slope
Equations that describe a line
y=mx+b
True
A number that tells how steeply a line slants as it goes up o…
What are linear equations
Equations that describe a line
Slope intercept form
y=mx+b
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
Linear Algebra (MATH 125) Exam 3
2 important observations regarding Mar…
Transition Matrix
If we can model an example as a Markov…
Markov Chain
1.) The market distribution at any specific time depends sole…
A matrix whose entries represent the probability of moving fr…
Sn = T^n * So... Sn = state of the system at time "n"... So = the…
A process in which the probability of a system being in a par…
2 important observations regarding Mar…
1.) The market distribution at any specific time depends sole…
Transition Matrix
A matrix whose entries represent the probability of moving fr…
Linear Algebra Math 129 Chapter 3
3.1: a real number that is useful in t…
3.1 DEF 1: the _________ of a 2x2 matr…
3.1 DEF 2: if A is a square atrix, the…
3.1 DEF 3: If A is a ___________ of or…
-determinant
-determinant
-minor (Mij)... -determinant... -cofactor (Cij)... -Cij=(-1)^i+j(Mij)
-square matrix... -determinant... -inductive... -expanding by cofactor…
3.1: a real number that is useful in t…
-determinant
3.1 DEF 1: the _________ of a 2x2 matr…
-determinant
Terms for Math 125: Linear Algebra For Business
linear equality
linear inequality
half plane
system of linear inequalities
mathematical expression in which all variables appear to the…
linear expression containing an inequality sign rather than a…
those points (x,y) for which the inequality is true (all poin…
collection of more than one linear inequality
linear equality
mathematical expression in which all variables appear to the…
linear inequality
linear expression containing an inequality sign rather than a…
5 terms
Linear Algebra (Math 313) Exam 3
Dimension of a Subspace
The dimensions of Nul A
The dimensions of Col A
Same Column Space
The number of vectors in the basis (all of the vectors in a s…
The number of free variables in the equation Ax=0
The number of pivot columns in A
...
Dimension of a Subspace
The number of vectors in the basis (all of the vectors in a s…
The dimensions of Nul A
The number of free variables in the equation Ax=0
Linear Algebra (MATH 125) Exam 2
Vectors provide what information?
Vector addition
Vector subtraction
Scalar multiplication
Magnitude (length) and direction
Done component-wise (is commutative*)... Vector v = <v1, v2, ..…
= one vector added to (-1)*another vector... Vector k = <-3, 2,…
Add the vector to itself however many times as indicated... For…
Vectors provide what information?
Magnitude (length) and direction
Vector addition
Done component-wise (is commutative*)... Vector v = <v1, v2, ..…
32 terms
Linear Algebra MATH0540 Melody Chan
invariant subspace
eigenvalue
Equivalent conditions to be an eigenva…
The restriction operator
Suppose T in L(V). A subspace U of V is called invariant unde…
A number lambda in F is called an eigenvalue of T if... there ex…
(a) Lambda is an eigenvalue of T ;... (b) T - lambda I is not in…
T |U in L(U) is defined by... T |U (u) = Tu... for u in U
invariant subspace
Suppose T in L(V). A subspace U of V is called invariant unde…
eigenvalue
A number lambda in F is called an eigenvalue of T if... there ex…
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…
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
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.
6 terms
HagenMath - Algebra ~ Systems of Linear Equations
elimination method
equivalent equations
infinitely many solutions
no solution
solves a system of equations by eliminating a variable by add…
equations that have the same solution(s)
linear systems of equations that have an infinite number of s…
when a system of equations fo have an intersection
elimination method
solves a system of equations by eliminating a variable by add…
equivalent equations
equations that have the same solution(s)
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…
65 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…
Linear Algebra Math 129 Chapters 1 and 2
In n variables; x1, x2, x3,....,xn has…
In in variables, is a set of m equatio…
1.1 DEF 1b: In this equation, a1, a2,.…
1.1 DEF 2a: A _______ of a linear equa…
linear equation
system of equations
-coefficients... -leading coefficients... -leading variable
-solution
In n variables; x1, x2, x3,....,xn has…
linear equation
In in variables, is a set of m equatio…
system of equations
46 terms
[MATH 110] Linear Algebra: Theorems, Corollaries, and Lemma
8 Conditions of vector space
Theorem 1.1: Cancellation Law for Vector
Corollary 1.1: Uniqueness of zero vector
Corollary 1.2: Uniqueness of additive…
P1: Closed under addition and scalar multiplication... 1. Additi…
Let V be vector space, if x, y, z are vectors in V such that…
The zero vector of a vector space is unique
The additive inverse of a vector in a vector space is unique
8 Conditions of vector space
P1: Closed under addition and scalar multiplication... 1. Additi…
Theorem 1.1: Cancellation Law for Vector
Let V be vector space, if x, y, z are vectors in V such that…
31 terms
Algebra 1 Linear Review
D
A
D
B
D
A
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.)
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…
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…
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
21 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…
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.…
17 terms
[MATH 217: Linear Algebra] Week 1
algebra
inconsistent system
matrix
square matrix
restoration (of broken parts)
a system with no solutions
rectangular array of numbers
a matrix in which the number of columns is equal to the numbe…
algebra
restoration (of broken parts)
inconsistent system
a system with no solutions
9 terms
Linear Algebra
vector
consistent
inconsistent
unique
ordered list of numbers
solution exists
no solution
one solution
vector
ordered list of numbers
consistent
solution exists
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