Study sets matching "linear algebra 2"

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Study sets matching "linear algebra 2"

12 terms
Linear Programming - Algebra 2
linear programming
feasible region
constraint
objective function
process of finding the minimum and maximum value of a functio…
the area of intersection of a system of inequalities
a limit or boundary of a region
the expression that defines the quantity to be maximized or m…
linear programming
process of finding the minimum and maximum value of a functio…
feasible region
the area of intersection of a system of inequalities
10 terms
Algebra 2 - Linear Equations
Linear Function
Linear Equation
Dependent Variable
Independent Variable
Function whose graph is a line.
Represents linear functions.
Y is called this.
X is called this.
Linear Function
Function whose graph is a line.
Linear Equation
Represents linear functions.
28 terms
Linear Algebra Test 2
Vector Space
If a system is inconsistent, is there…
Span of a vector space (set of vectors…
Linearly dependent
A linear combination of v1...vn with coefficients c1...cn is…
No
The set of all possible linear combinations with vectors in t…
At least one vector in S can be written as a linear combinati…
Vector Space
A linear combination of v1...vn with coefficients c1...cn is…
If a system is inconsistent, is there…
No
9 terms
Algebra Linear functions 2
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…
22 terms
Algebra 2 Unit 4 Linear Models
Scatter Plot
Correlation
Trend Line
Line of Best Fit
a graph that relates two sets of data by plotting the data as…
relationship between data sets
a line that approximates the relationship between variables o…
a trend line that gives the most accurate model of related data
Scatter Plot
a graph that relates two sets of data by plotting the data as…
Correlation
relationship between data sets
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…
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...
Algebra 2 - Linear Equations
Linear Function
Linear Equation
Dependent Variable
Independent Variable
Function whose graph is a line.
Represents linear functions.
Y is called this.
X is called this.
Linear Function
Function whose graph is a line.
Linear Equation
Represents linear functions.
Linear Algebra Chapter 2
Definition of a subspace
Column Space
Null Space
Theorem 12 for subspaces
A subspace of R^n is any set H in R^n that has three properti…
The column space of a matrix A is the set Col A of all linear…
The null space of a matrix A is the set Nul A of all solution…
The null space of an mxn matrix A is a subspace of R^n. Equiv…
Definition of a subspace
A subspace of R^n is any set H in R^n that has three properti…
Column Space
The column space of a matrix A is the set Col A of all linear…
Linear Algebra 1.2
Reduced Echelon Form
Echelon Form
Neither
...
...
...
Reduced Echelon Form
...
Echelon Form
...
8 terms
Algebra 2: Linear Programming
system of linear inequalities
solutions of a system
half plane
Optimization
Graph of two linear inequalities using the same variables.
Any ordered pairs (x,y) that makes both inequalities true.
the part of a plane on one side of an indefinitely extended s…
A process in which the minimum or maximum of a quantity is so…
system of linear inequalities
Graph of two linear inequalities using the same variables.
solutions of a system
Any ordered pairs (x,y) that makes both inequalities true.
22 terms
Linear Algebra Quiz 2
Linear Equation
System of Linear Equations
Solution of a System
Consistent System
In the variables x1, x2, x3, ...., xn is an equation that can…
A collection of one or more linear equations involving the sa…
a list (s1, s2, ..., sn) of numbers that makes each equation…
A system that has at least one solution
Linear Equation
In the variables x1, x2, x3, ...., xn is an equation that can…
System of Linear Equations
A collection of one or more linear equations involving the sa…
7 terms
Linear Algebra Exam 2
If A is axb and B is cxd, AB has the d…
If two rows of A are interchanged to p…
If one row of A is multiplied by a con…
Found by finding all pivot positions a…
axd (# rows A x # columns B)
detB = - detA
detB = (k)detA
basis for Col(A)
If A is axb and B is cxd, AB has the d…
axd (# rows A x # columns B)
If two rows of A are interchanged to p…
detB = - detA
Linear Algebra Midterm 2
For an m x n matrix, A, the rank of A…
What is the rank of A^T, the transpose…
True or False: If A is an m x n matrix…
What is the nullity of a 5 x 4 matrix…
The number of leading ones in reduced Echelon form.
The rank of A
True
N(A) = n - R(A) = 4 - 3 = 1
For an m x n matrix, A, the rank of A…
The number of leading ones in reduced Echelon form.
What is the rank of A^T, the transpose…
The rank of A
16 terms
Chapter 2 Linear Algebra
Diagonal Matrix
Matrix Addition
Addition Properties
When multiplying a matrix by a constant,
square nxn matrix with nonzero diagonal entries and zeros in…
add corresponding entries, matrices must be same size
a) A+B = B+A... b) (A+B)+C = A+(B+C)... c) A+0 = A... d) r(A+B) = rA+r…
all entries are multiplied by the constant
Diagonal Matrix
square nxn matrix with nonzero diagonal entries and zeros in…
Matrix Addition
add corresponding entries, matrices must be same size
Algebra 2: Graphing Linear Relationships
y > 2x + 3
y < 2x +3
y > -x +5
y < -x +5
y > 2x + 3
y < 2x +3
18 terms
Linear Algebra Exam 2
standard matrix
onto
one-to-one
zero matrix
For a linear transformation T:R^n->R^m, there exists a unique…
A transformation T:R^n->R^m is onto if every vector b in R^m…
A transformation T:R^n->R^m is one-to-one if every vector b i…
a matrix containing only zeros
standard matrix
For a linear transformation T:R^n->R^m, there exists a unique…
onto
A transformation T:R^n->R^m is onto if every vector b in R^m…
10 terms
Linear Algebra 2.2
Commutative Property of Addition
Associative Property of Additon
Associative Property of Multiplication
Multiplicative Identity
A + B = B + A
(A + B) + C = A + (B + C)
(cd) A = c (dA)
1(A) = A
Commutative Property of Addition
A + B = B + A
Associative Property of Additon
(A + B) + C = A + (B + C)
Algebra 2 linear equations
Horizontal shifts
Vertical Shifts
Reflection across Y axis
Reflection across x axis
f(x-h)
f(x)+k
f(-x)
-f(x)
Horizontal shifts
f(x-h)
Vertical Shifts
f(x)+k
21 terms
Linear Algebra 2.6
Similar matrices
Similarity transformation of matrices
Diagonalizable matrix
Diagonalization Theorem.
Matrices A and B such that P^-1AP = B or equivalently A = PBP…
it is when changing A into PDP^-1
A matrix that can be written in factored form as PDP^-1, wher…
An matrix A is diagonalizable if and only if A has n linearly…
Similar matrices
Matrices A and B such that P^-1AP = B or equivalently A = PBP…
Similarity transformation of matrices
it is when changing A into PDP^-1
Linear Algebra 2
Eigenvalues
eigenvectors
Diagonalization
Unit vector
a-j across diagonal, solve for zero
free variable in parametric form (basically null of eigenspace)
P is eigenvectors together, D is diagonal matrix with eigenva…
sqrt of sum of squares is one
Eigenvalues
a-j across diagonal, solve for zero
eigenvectors
free variable in parametric form (basically null of eigenspace)
Linear Algebra Exam 2
Column Space
Null Space
Subspace
Basis for a Subspace
colA is the set of all possible linear combinations of column…
nulA is a set of solutions to Ax=0, Dependent columns
1. 0 vector is in H... 2. If u and v are in H, then u + v is in…
Any set of linearly independent vectors (not unique)
Column Space
colA is the set of all possible linear combinations of column…
Null Space
nulA is a set of solutions to Ax=0, Dependent columns
19 terms
Linear Algebra 2
A Square Matrix that has no inverse.
An n x n matrix A such that if there i…
An m x n matrix A, whose turned into a…
A square n x n matrix whose non diagon…
Singular (Matrix)
Invertible
Transpose
Diagonal Matrix
A Square Matrix that has no inverse.
Singular (Matrix)
An n x n matrix A such that if there i…
Invertible
Algebra 2 Linear Equations
Inconsistent
Consistent and independent
Consistent and dependent
Horizontal
If the slopes are the same but the y-intercepts are different
If the slopes are different
If the slopes are the same and the y-intercepts are the same
Y term
Inconsistent
If the slopes are the same but the y-intercepts are different
Consistent and independent
If the slopes are different
9 terms
Linear Algebra Terms (2)
Null Space
Column Space
Row Space
Left Null Space
the set of all vectors x that are solutions to the homogenous…
the set of all possible linear combinations of the matrix's c…
the column space the transpose of A; the set of all linear co…
the left null space is the null space of the transpose of A;…
Null Space
the set of all vectors x that are solutions to the homogenous…
Column Space
the set of all possible linear combinations of the matrix's c…
10 terms
Linear Algebra 2.1
Equal Matrices
Column matrix
Row matrix
Sum
If matrices have the same size
(Column vector) A matrix with only one column
(Row vector) A matrix with only one row
If matrices A and B are the same size mxn they can be added.
Equal Matrices
If matrices have the same size
Column matrix
(Column vector) A matrix with only one column
16 terms
Linear Algebra 2
orthogonality, length, unit vectors
orthonormal vectors
properties of orthonormal vectors
orthogonal projection
.
.
.
.
orthogonality, length, unit vectors
.
orthonormal vectors
.
21 terms
Linear Algebra Def. 2
Partitioned (or block) matrix
Column--row expansion of AB
LU factorization
Solving the equation LUx=b
A matrix whose entries are themselves matrices of appropriate…
The expression of a product AB as a sum of outer products: co…
The representation of a matrix A in the form A = LU where L i…
L is the lower triangular form and U is the upper triangular…
Partitioned (or block) matrix
A matrix whose entries are themselves matrices of appropriate…
Column--row expansion of AB
The expression of a product AB as a sum of outer products: co…
Linear Algebra 2
Define:... A Binary Operation
Define:... A Group
Define:... Abelian
Define:... Finite
A binary operation on a set G is a map G x G -> G from the Ca…
A group is a set G together with a binary operation G x G ->…
A group is called abelian if for every a and b in G, a.b = b.a
A group is called finite if it has finitely many elements.
Define:... A Binary Operation
A binary operation on a set G is a map G x G -> G from the Ca…
Define:... A Group
A group is a set G together with a binary operation G x G ->…
26 terms
Linear Algebra Theorems 2
The Invertible Matrix Theorem
Let A and B be square matrices. If AB…
Let T be a linear transformation (R^n…
Column-Row expansion of AB
Let A be an nxn matrix. Then the following are equivalent:... a.…
A and B are both invertible, with B=A^-1 and A=B^-1
A is invertible
If A is mxn and B is nxp, then... AB=col1(A)row1(B)+...+coln(A)…
The Invertible Matrix Theorem
Let A be an nxn matrix. Then the following are equivalent:... a.…
Let A and B be square matrices. If AB…
A and B are both invertible, with B=A^-1 and A=B^-1
12 terms
Linear Programming - Algebra 2
linear programming
feasible region
constraint
objective function
process of finding the minimum and maximum value of a functio…
the area of intersection of a system of inequalities
a limit or boundary of a region
the expression that defines the quantity to be maximized or m…
linear programming
process of finding the minimum and maximum value of a functio…
feasible region
the area of intersection of a system of inequalities
9 terms
6.2 Linear Algebra
A set of vectors is an orthogonal set…
If an orthogonal set consists of nonze…
An orthogonal basis for a subspace W o…
An m x m matrix U has orthonormal colu…
each pair of distinct vectors from the set is orthogonal
linearly independent and a basis for the subspace spanned by…
orthogonal set
U^TU = I
A set of vectors is an orthogonal set…
each pair of distinct vectors from the set is orthogonal
If an orthogonal set consists of nonze…
linearly independent and a basis for the subspace spanned by…
14 terms
Linear Algebra 1.2
Leading entry
Rectangular matrix is in Row echelon f…
Rectangular matrix is in reduced Row e…
Row reduction
Leftmost nonzero entry in a row
1. All nonzero rows are above any rows of all zeroes... 2. Each…
In addition to the matrix being in echelon form:... 4. The lead…
Transformation of s matrix by elementary row operations
Leading entry
Leftmost nonzero entry in a row
Rectangular matrix is in Row echelon f…
1. All nonzero rows are above any rows of all zeroes... 2. Each…
Linear Algebra Chapter 2.2
In order for a matrix B to be an inver…
If A and B are n x n and invertible, t…
If A = [a b] and ab - cd does ... [c d]…
If A is an invertible n x n matrix, th…
True.
False. The inverse of two invertible matrices is the reverse…
False. Then the statement would be contrapositive to the stat…
True
In order for a matrix B to be an inver…
True.
If A and B are n x n and invertible, t…
False. The inverse of two invertible matrices is the reverse…
14 terms
Algebra (Chap 2) - Linear Equations
Formula
Solve an Equation
Equivalent Equations
Multi-Step Equations
A mathematical relationship or rule expressed in symbols.
The process of finding all values of the variable that make t…
equations that have the same solution
Equations with more than one operation.
Formula
A mathematical relationship or rule expressed in symbols.
Solve an Equation
The process of finding all values of the variable that make t…
26 terms
Linear Algebra 2.2
Determinant of n×n matrix.
(i,j)-cofactor of a matrix A.
Cofactor expansion across the first ro…
Determinant of a triangular matrix.
The number det A defined inductively by a cofactor expansion…
is the number Cij given by Cij = (-1)^(i+j) det Aij then det…
det A = a11C11 + a12C12 + ... + A1nC1n
the det A is the product of the entries on the main diagonal…
Determinant of n×n matrix.
The number det A defined inductively by a cofactor expansion…
(i,j)-cofactor of a matrix A.
is the number Cij given by Cij = (-1)^(i+j) det Aij then det…
58 terms
Linear algebra test #2
Invertible matrix A
2x2 A-1
IMT key points
IMT later points
A*A-1 = In (and the other way must be true as well)
1/(ad-bc)* swap a and d, reverse signs on b and c
Row equivalent to In, transformation using it as standard mat…
DetA is not 0, columns of A form A basis for Rn (linearly ind…
Invertible matrix A
A*A-1 = In (and the other way must be true as well)
2x2 A-1
1/(ad-bc)* swap a and d, reverse signs on b and c
Linear Algebra Quiz #2
linear combination
the span
to span
product of Ax
. GIven vectors v1,v2...vn in R^n and scalars c1,c2...cn y={c…
The set of all linear combinations of v1.v2...vn
Span{v1,v2...vn}={c1v1+c2v2+...cnvn} such that c1,c2...cn are…
Linear combination of the columns of A with the weights comin…
linear combination
. GIven vectors v1,v2...vn in R^n and scalars c1,c2...cn y={c…
the span
The set of all linear combinations of v1.v2...vn
90 terms
Linear Algebra Exam #2
Matrix factorization:... characteristics…
Solving an LU factorization:
Purpose of the LU factorization... (and c…
Leontief Input-Output Model (Productio…
A: m x n matrix that can be row reduced to echelon form (w/o…
solve the pair of equations, in order:... Ly=b for y... Ux=y for…
provides a more efficient method of solving a sequence of equ…
x = Cx + d... where:... x is a vector representing amount produced…
Matrix factorization:... characteristics…
A: m x n matrix that can be row reduced to echelon form (w/o…
Solving an LU factorization:
solve the pair of equations, in order:... Ly=b for y... Ux=y for…
20 terms
Linear Algebra Test #2
Linearly Independent
Symmetric Matrix
(kA)^T
(A^r)^T
There is no combination of scalars that you can multiply with…
A transpose is equal to A
k(A^T)
(A^T)^r
Linearly Independent
There is no combination of scalars that you can multiply with…
Symmetric Matrix
A transpose is equal to A
23 terms
Linear Algebra Chp 2
Matrix Addition
Scalar Multiplication
Isomorphism
Standard Basis Vectors
Let A and B both be mxn matrices. Then the matrix A+B is the…
Let A be an mxn matrix and let c be a scalar. Then the matrix…
The behavior that when two elements of R^n are added as n-tup…
Vectors ei for i=1,...,n in R^n... Ex:... In R^1 we have one of the…
Matrix Addition
Let A and B both be mxn matrices. Then the matrix A+B is the…
Scalar Multiplication
Let A be an mxn matrix and let c be a scalar. Then the matrix…
Linear Algebra Set 2
Coordinate vector with respect to a ba…
Product of a matrix and a vector
Product of an l × m matrix B and an m…
Null space of a matrix
Let V be a subspace of R... n, B = (v1, . . . , vk) a basis of V…
Let A be an m × n matrix with columns a1, . . . , an and let…
Let the columns of A be a1, . . . , an. Then BA is the l × n…
Let A be an m × n matrix. Then the null space of A, denoted b…
Coordinate vector with respect to a ba…
Let V be a subspace of R... n, B = (v1, . . . , vk) a basis of V…
Product of a matrix and a vector
Let A be an m × n matrix with columns a1, . . . , an and let…
30 terms
Linear Algebra Chapter 2
dimension
Rn
normalized n-tuple
A1
number of elements (columns) in a row matrix, number of vecto…
Set of ordered arrays of n real numbers. The set is represent…
one thats magnitude = 1 (divide all the original numbers by t…
Closure under addition: if u and v belong to V, then so does…
dimension
number of elements (columns) in a row matrix, number of vecto…
Rn
Set of ordered arrays of n real numbers. The set is represent…
29 terms
Linear algebra test #2
If A and B are 2x2 matrices with colum…
Each column of AB is a linear combinat…
AB+AC=A(B+C)
A^T+B^T=(A+B)^T
False. Matrix multiplication is row by column
False. Swap A and B, then it's true.
True. Matrix multiplication distributes over addition
True. Properties of transportation. Also when we add we add c…
If A and B are 2x2 matrices with colum…
False. Matrix multiplication is row by column
Each column of AB is a linear combinat…
False. Swap A and B, then it's true.
10 terms
Linear Algebra Chapter 2
Theorem 1
Theorem 2
Theorem 3
Theorem 4
Let A,B, and C be matrices of the same size, and let r and s…
Let A be an mX n matrix and let B and C have sizes for which…
Let A and B denote matrices whose sizes are appropriate for t…
Let A equal a matrix. If ad- bc ~=0 then A is invertible and…
Theorem 1
Let A,B, and C be matrices of the same size, and let r and s…
Theorem 2
Let A be an mX n matrix and let B and C have sizes for which…
Linear Algebra Chapter 5.2
Similar
the Characteristic Equation
The determinant of A is the product of…
An elementary row operation on A does…
two matrices are similar if there exists an invertible matrix…
det(A - YI) = 0 (the characteristic equation of A)
False, true only when the matrix is triangular
False, interchanging rows or multiplying row with a constant…
Similar
two matrices are similar if there exists an invertible matrix…
the Characteristic Equation
det(A - YI) = 0 (the characteristic equation of A)
11 terms
Linear Algebra Test #2
Isomorphism satisfies which 3 conditions
What does 1 to 1 mean?
What does Onto mean?
How do you test that it preserves stru…
Preserve Structure... 1-1... Onto
No two inputs have the same output
Every y is used. Every output is used.
a) f(V1+V2) in v = f(V1) + f(V2) in w... b) f(rv) in v = rf(v…
Isomorphism satisfies which 3 conditions
Preserve Structure... 1-1... Onto
What does 1 to 1 mean?
No two inputs have the same output
34 terms
Linear Algebra Chapter 2
Matrix Addition
Scalar multiplcation
A product AB of matrices can be formed…
pg 59... The ij entry of AB is formed fro…
Let A and B both be mxn matrices. Then the matrix A + B is th…
Let A be an mxn matrix and let c be a scalar. Then the matrix…
Fill in the blank:... 1. rows... 2. columns... 3. rows... 4. columns
Fill in the blank... 1. row i... 2. column j
Matrix Addition
Let A and B both be mxn matrices. Then the matrix A + B is th…
Scalar multiplcation
Let A be an mxn matrix and let c be a scalar. Then the matrix…
Linear Algebra Chapter 7.2
a. The matrix of a quadratic form is a…
b. A quadratic form has no cross-produ…
c. The principal axes of a quadratic f…
d. A positive definite quadratic form…
True
True
True
False, Q(x) = 0 when x = 0
a. The matrix of a quadratic form is a…
True
b. A quadratic form has no cross-produ…
True
Linear Algebra Chapter 3.2
A row replacement operation does not a…
The determinant of A is the product of…
If the columns of A are linearly depen…
det( A + B) = det A + det B
True. Thm 3 Part A
True
True. If the columns of A are Linearly Dependent, then by the…
False
A row replacement operation does not a…
True. Thm 3 Part A
The determinant of A is the product of…
True
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