# Study sets matching "linear algebra 2"

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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

orthogonal complement of a vector space

4 Fundamental Spaces

TA

domain

The set of all vectors orthogonal to all the vectors in a vect…

row(A) col(A) null(A) null(At)

matrix transformation

all possible inputs

orthogonal complement of a vector space

The set of all vectors orthogonal to all the vectors in a vect…

4 Fundamental Spaces

row(A) col(A) null(A) null(At)

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

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.

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

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"…

Additive Inverse of a Matrix (2.2)

Zero Matrix (2.2)

Identity Matrix (2.2)

Transpose of a Matrix (2.2)

The Zero matrix of 'm x n' (Let's say m x n is = to 2 x 3) is…

If A is 'm x n' matrix and O_mn is the 'm x n' matrix consisti…

The Matrix I serves as the identity for Matrix multiplication:…

Is formed by writing a matrices' columns as rows and rows as c…

Additive Inverse of a Matrix (2.2)

The Zero matrix of 'm x n' (Let's say m x n is = to 2 x 3) is…

Zero Matrix (2.2)

If A is 'm x n' matrix and O_mn is the 'm x n' matrix consisti…

(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...

diagonal matrix

identity matrix

zero matrix

transpose

A diagonal matrix is a square matrix whose only nonzero entrie…

The n × n identity matrix In is the di- agonal matrix with all…

The zero matrix (of size m×n) is the 0 m×n matrix 0 with all z…

The transpose of an m × n matrix A... is the n×m matrix AT whose…

diagonal matrix

A diagonal matrix is a square matrix whose only nonzero entrie…

identity matrix

The n × n identity matrix In is the di- agonal matrix with all…

NulA

ColA

Determinant

Basis

for n x n matrix A is the set of all vectors x(bar) so that Ax…

1) the set of all linear combinations of the columns of A... 2)…

Let A be an n x n matrix, the determinant is the sum from i =…

Let H be a subspace of R^n, a basis for H is a set of vectors…

NulA

for n x n matrix A is the set of all vectors x(bar) so that Ax…

ColA

1) the set of all linear combinations of the columns of A... 2)…

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.

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…

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

Determinant of n×n matrix.

(i,j)-cofactor of a matrix A.

Cofactor expansion across the first row…

Determinant of a triangular matrix.

The number det A defined inductively by a cofactor expansion a…

is the number Cij given by Cij = (-1)^(i+j) det Aij then det A…

det A = a11C11 + a12C12 + ... + A1nC1n

the det A is the product of the entries on the main diagonal o…

Determinant of n×n matrix.

The number det A defined inductively by a cofactor expansion a…

(i,j)-cofactor of a matrix A.

is the number Cij given by Cij = (-1)^(i+j) det Aij then det A…

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)