#### Study sets matching "linear algebra 2"

#### Study sets matching "linear 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

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

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…

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

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…

(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...

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…

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.

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…

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

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

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

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…

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)

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

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)

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

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

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

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…

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

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…

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 ->…

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

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

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…

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…

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…

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…

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…

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 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

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…

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…

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…

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…

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.

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…

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)

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(r

**v) in v = r**f(v…Isomorphism satisfies which 3 conditions

Preserve Structure... 1-1... Onto

What does 1 to 1 mean?

No two inputs have the same output

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…

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

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