#### 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 function…

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 mi…

linear programming

process of finding the minimum and maximum value of a function…

feasible region

the area of intersection of a system of inequalities

Augmented Matrix

3x4 Matrix

Elementary Row Operations for Matrices

Row Equivalent Matrices

3 rows and 4 columns

(1) Interchange any two rows... (2) Multiply a row by any nonzero…

Matrices that have a sequence of row operations to transform o…

Augmented Matrix

3x4 Matrix

3 rows and 4 columns

Zero Matrix

-A in A+(-A)=0 is called the...

Diagonal Matrix

Identity Matrix

Matrix with all zeros

additive inverse of A

a square matrix whose non-diagonal entries are zeros

a square matrix with ones on the main diagonal and zeros every…

Zero Matrix

Matrix with all zeros

-A in A+(-A)=0 is called the...

additive inverse of A

Equivalent Matrices have the same...

Column Matrix/Column Vector

Row Matrix/Row Vector

Size of the product of mxn and nxp is

size and entries

Matrix with only one column

Matrix with only one row

mxp

Equivalent Matrices have the same...

size and entries

Column Matrix/Column Vector

Matrix with only one column

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 or…

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 Ec…

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. E…

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 propertie…

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 solutions…

The null space of an mxn matrix A is a subspace of R^n. Equiva…

Definition of a subspace

A subspace of R^n is any set H in R^n that has three propertie…

Column Space

The column space of a matrix A is the set Col A of all linear…

Coordinate Plane

x-axis

y-axis

Origin

Formed by two perpendicular real number lines.

The horizontal line of the coordinate plane.

The vertical line of the coordinate plane.

The point at which the two lines meet.

Coordinate Plane

Formed by two perpendicular real number lines.

x-axis

The horizontal line of the coordinate plane.

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 sam…

a list (s1, s2, ..., sn) of numbers that makes each equation i…

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 sam…

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 st…

A process in which the minimum or maximum of a quantity is sou…

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.

If A is axb and B is cxd, AB has the di…

If two rows of A are interchanged to pr…

If one row of A is multiplied by a cons…

Found by finding all pivot positions an…

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 di…

axd (# rows A x # columns B)

If two rows of A are interchanged to pr…

detB = - detA

For an m x n matrix, A, the rank of A i…

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 w…

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 i…

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 n…

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+rB…

all entries are multiplied by the constant

Diagonal Matrix

square nxn matrix with nonzero diagonal entries and zeros in n…

Matrix Addition

add corresponding entries, matrices must be same size

True

False - A has 2 rows

True

False - the cancellation law does not a…

If A and B are mxn, then both AB^T and A^TB are defined

If AB=C and C has 2 columns, then A has 2 columns

Left-multiplying a matrix B has a diagonal matrix A, with nonz…

If BC = BD, then C = D

True

If A and B are mxn, then both AB^T and A^TB are defined

False - A has 2 rows

If AB=C and C has 2 columns, then A has 2 columns

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 i…

A transformation T:R^n->R^m is one-to-one if every vector b is…

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 i…

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)

let A, B, C, D calculate each of the fo…

find the standard matrix for the projec…

given four column vectors check that ac…

Suppose A and B are symmetric. Show tha…

first column becomes first row

A= 1/ length^2 times a times a transposed = A

product is a linear combo of the rows of the second matrix

A=A^T and B=B^T. prove (AB)^T = BA. for (AB)^T to be symmetric…

let A, B, C, D calculate each of the fo…

first column becomes first row

find the standard matrix for the projec…

A= 1/ length^2 times a times a transposed = A

Echelon form

Reduced echelon form

Row reducing

Pivot position

Nonzero rows above zero rows, each leading entry is to the rig…

An echelon form matrix that has 1's for leading entries, that…

Transformation of a nonzero matrix into echelon form

Position of a leading 1 of a reduced echelon form matrix

Echelon form

Nonzero rows above zero rows, each leading entry is to the rig…

Reduced echelon form

An echelon form matrix that has 1's for leading entries, that…

Nul A

Col A

Basis of a subspace of H of a vector sp…

Prove H is a subspace of V

The nullspace of a matrix A is the set Nul A of all solutions…

The columnspace of a matrix A is the set Col A of all linear c…

A basis for a subspace H of a vector space v is a linearly ind…

H is a subset of V... a) The zero vector of V is in H... b) H is c…

Nul A

The nullspace of a matrix A is the set Nul A of all solutions…

Col A

The columnspace of a matrix A is the set Col A of all linear c…