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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.
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...
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…
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
27 terms
Algebra 2: Graphing Linear Relationships
y > 2x + 3
y < 2x +3
y > -x +5
y < -x +5
y > 2x + 3
y < 2x +3
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
11 terms
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…
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.
10 terms
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…
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…
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…
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.
15 terms
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
17 terms
Linear Algebra 1.2
Reduced Echelon Form
Echelon Form
Neither
...
...
...
Reduced Echelon Form
...
Echelon Form
...
41 terms
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
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…
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
10 terms
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)
8 terms
Algebra 2 linear functions
Horizontal shift (#) right
Horizontal shift (#) left
Vertical Shift (#) up
Vertical shift (#) down
f(x-#)
f(x+#)
f(x)+#
f(x) - #
Horizontal shift (#) right
f(x-#)
Horizontal shift (#) left
f(x+#)
18 terms
Linear Algebra Definitions 2
ordered basis
coordinate vector of x relative to ord…
A*B, where A and B are mxn and nxp mat…
L_A
basis with specific order, for Rn it's {e_1, ... , e_n}, and…
[x]_beta = column version of (a1,a2,...,an)
(AB)_ij = (sum from k = 1 to n) A_ik * B_kj, 1 <= i <= m, 1 <…
L_A: Rn --> Rm, where L_A(x) = A*x
ordered basis
basis with specific order, for Rn it's {e_1, ... , e_n}, and…
coordinate vector of x relative to ord…
[x]_beta = column version of (a1,a2,...,an)
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…
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
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…
6.2 Linear Algebra
T(u)=Au; nonsingular
lines; segments of lines; parallel lin…
||u|| = ||T(u)||; equal to; d(P, Q)=d(…
T(u)=u+v
A nonsingular transformation is a mapping defined by ________…
A nonsingular linear transformation T maps... (a) lines into ___…
Let T be an orthogonal transformation on R^n. Let u and v be…
A Translation is a transformation T:Rn... to Rm defined by _____…
T(u)=Au; nonsingular
A nonsingular transformation is a mapping defined by ________…
lines; segments of lines; parallel lin…
A nonsingular linear transformation T maps... (a) lines into ___…
36 terms
Linear Algebra Test #2
Transformation T from Rⁿ to Rm
Linear Transformation
Theorem 10 (Ch. 1)
Standard matrix for the linear transfo…
a rule that assigns to each vector *x* in Rⁿ a vector T(*x*)…
A transformation is linear if ... (i) T(*u*+*v*) = T(*u*) + T(*v…
Let T: Rⁿ→Rm be a linear transformation. Then there exists a…
The matrix in A = [T(e₁) ... T(en)].
Transformation T from Rⁿ to Rm
a rule that assigns to each vector *x* in Rⁿ a vector T(*x*)…
Linear Transformation
A transformation is linear if ... (i) T(*u*+*v*) = T(*u*) + T(*v…
49 terms
linear algebra 2
determinant properties (4)
if a has a zero column
if a has two equal columns
if one column of a is a multiple of an…
linearity, no change w/ column replacement, antisymmetry, norm
det(A)=0
det(A)=0
det(A)=0
determinant properties (4)
linearity, no change w/ column replacement, antisymmetry, norm
if a has a zero column
det(A)=0
30 terms
Linear Algebra 2.7
Orthogonal projection
Properties of matrix with orthonormal…
Orthogonal matrix.
Orthogonal Decomposition Theorem.
y.u/u.u u
1. An mxn matrix U has orthonormal columns if and only if U^T…
a Square invertible matrix U such that U^-1 = U^T
^y = (y.u1/u.u) u1 + .... (y.up/up/up) up
Orthogonal projection
y.u/u.u u
Properties of matrix with orthonormal…
1. An mxn matrix U has orthonormal columns if and only if U^T…
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…
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
16 terms
Linear Algebra 2
orthogonality, length, unit vectors
orthonormal vectors
properties of orthonormal vectors
orthogonal projection
.
.
.
.
orthogonality, length, unit vectors
.
orthonormal vectors
.
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…
9 terms
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
9 terms
Linear Algebra Exam #2
How to find if something is a vector s…
How to find if something is a subspace:
To find if a set of vectors are a span…
Row Echelon Form Requirements:
Follows the 8 axioms and is closed for scalar multiplication…
Show it is closed for scalar multiplication and addition:... α(x…
1. This means that a constant times each of vectors will equa…
1. The first entry in each row is 1... 2. For any row below it,…
How to find if something is a vector s…
Follows the 8 axioms and is closed for scalar multiplication…
How to find if something is a subspace:
Show it is closed for scalar multiplication and addition:... α(x…
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.
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
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…
19 terms
2 Linear Algebra
Scalar Product, Dot Product, Inner Pro…
Dot product rules
Cauchy-Schwarz Inequality
Perpendicular
Legnth of vector x multiplied by the length of vector y multi…
Order doesnt matter, distributive accross addition, can take…
Length of the dot product of x and y is < length of x multipl…
Orthogonal, normal. 0 Dot product
Scalar Product, Dot Product, Inner Pro…
Legnth of vector x multiplied by the length of vector y multi…
Dot product rules
Order doesnt matter, distributive accross addition, can take…
15 terms
Linear Algebra - Chapter 2
solutions to systems of homogeneous eq…
span
How to determine if a vector is in a s…
Any 2 vectors that are NOT scalar mult…
subspaces
when a subspace of a vector space consists of all linear comb…
create a system of equations using the vector in question as…
will span all of R2
solutions to systems of homogeneous eq…
subspaces
span
when a subspace of a vector space consists of all linear comb…
26 terms
Linear Algebra Quiz 2
Linear equation (In the variables x1..…
System of linear equations
Solution set of the linear system
Equivalent linear systems
An equation that can be written in the form a1x1 +a2x2+...anx…
A collection of one or more linear equations involving the sa…
A list of numbers(s1...sn) that makes each equation in the sy…
Linear systems with the same solution set.
Linear equation (In the variables x1..…
An equation that can be written in the form a1x1 +a2x2+...anx…
System of linear equations
A collection of one or more linear equations involving the sa…
11 terms
Linear Algebra exam 2
Linear combination
Vector
Scalar
Parallelogram rule for addition
Doing both: adding two vectors and multiplying a scalar by a…
Matrix with only one column
Literally any real number: denoted as a vector with one entry
If u and v in R^2 are represented as points in the plane, the…
Linear combination
Doing both: adding two vectors and multiplying a scalar by a…
Vector
Matrix with only one column
11 terms
Linear Algebra Exam 2
Linear Independence
Linear Dependent
det|A|=0
det|A| doesn't equal 0
when the set of vectors has only the trivial solution
when the set of vectors have has solutions that are also nont…
there are infinite solutions, thus linearly dependent
only trivial solution exists, thus linearly independent
Linear Independence
when the set of vectors has only the trivial solution
Linear Dependent
when the set of vectors have has solutions that are also nont…
8 terms
Linear Algebra Quiz 2
Column spans matrix if...
indepedent
dependent
Homogenous Ax=0 has non-trivial soluti…
...if there's a pivot in every row
trivial solutions
non-trivial solutions
equation has 1+ free variables
Column spans matrix if...
...if there's a pivot in every row
indepedent
trivial solutions
10 terms
Linear Algebra Test 2
S=LL^t
A=LU
A=VΛV^-1
A=UΣV^t
Cholesky
LU decomposition
Eigen Decomposition
Single Value Decomposition
S=LL^t
Cholesky
A=LU
LU decomposition
linear algebra test 2
Prop 2.3.2
Prop 2.3.5
Prop 2.3.6
Prop 2.5.1
some rules of T
t and s are the same
you can find any linear transformation that works
range of T
Prop 2.3.2
some rules of T
Prop 2.3.5
t and s are the same
106 terms
Linear Algebra Test 2
Definition 3.3.1: subspace
Definition 3.3.2: span
Theorem 3.3.3
Theorem 3.3.4
- If a collection of vectors V in Rn is also a vector space u…
- Let X be a collection of vectors in Rn. The span of X is th…
- If X is a collection of vectors in Rn, then Span(X) is a su…
- If V is a collection of vectors in Rn such that 0 is in V a…
Definition 3.3.1: subspace
- If a collection of vectors V in Rn is also a vector space u…
Definition 3.3.2: span
- Let X be a collection of vectors in Rn. The span of X is th…
Linear Algebra Test 2
Diagonalizable Matrices
Eigenvectors
Eigenvalue
Eigenbasis
Matrix A is diagonalizable iff A is similar to some diagonal…
A nonzero vector v in Rⁿ is ____ of A if Av = λv for some sca…
λ of the associated eigenvector. We find it by setting det(A-…
A basis formed from the eigenvectors for A, meaning that Av₁…
Diagonalizable Matrices
Matrix A is diagonalizable iff A is similar to some diagonal…
Eigenvectors
A nonzero vector v in Rⁿ is ____ of A if Av = λv for some sca…
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