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Study sets matching "linear math algebra"

31 terms
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 )
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
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)
100 terms
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…
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…
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…
48 terms
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
43 terms
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
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.
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
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
11 terms
MATH 307 Applied Linear Algebra
Hilbert-Schmidt norm (for a matrix)
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)
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"…
30 terms
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
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…
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…
31 terms
Algebra 1 Linear Review
D
A
D
B
D
A
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…
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
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…
31 terms
Algebra 1 Linear Review
D
A
D
B
D
A
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.…
56 terms
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…
9 terms
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
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.
70 terms
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…
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
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…
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…
23 terms
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…
70 terms
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
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.
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…
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…
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…
14 terms
Algebra 1: Linear Equations
Rate of change
Slope
The Slope Formula
Positive Slope
Rate of change
Slope
16 terms
Math Unit 3 Algebraic Linear Equations
Algebraic equations
coefficient
consistent equation
constant
equation that includes one or more variables... 4x + 7 = 3x + 14
the number in front of the non-numerical symbols in an algebr…
an equation with only one equation
a term without a variable
Algebraic equations
equation that includes one or more variables... 4x + 7 = 3x + 14
coefficient
the number in front of the non-numerical symbols in an algebr…
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
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…
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
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…
70 terms
Algebra I; MyMathLab; Unit III; Linear Equations and Inequalities in One Variable
Equation
Translating Words to Equations:... When…
1.) Understand the problem.... 2.) Choose…
Variable
is a mathematical statement that two expressions are equal. A…
1.) Read the word problem carefully to get an overview.... 2.) D…
EXAMPLE:... One-third of a number is fourteen.... 1/3 x n = 14... 1/3…
is a letter or symbol that represents an unknown quantity.
Equation
is a mathematical statement that two expressions are equal. A…
Translating Words to Equations:... When…
1.) Read the word problem carefully to get an overview.... 2.) D…
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