# Study sets matching "linear math algebra"

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inconsistent

consistent

homogeneous

underspecified

a system of linear equations that has no solution

a system of linear equations that has a solution

if the matrix is empty and all the variables are 0 (or somethi…

more unknowns than equations

inconsistent

a system of linear equations that has no solution

consistent

a system of linear equations that has a solution

x bar parallel is

two vectors are orthogonal to each othe…

magnitude is

In an LU decomposition, the columns of…

the orthogonal projection of x bar onto the line

u * v = 0

the square root of each component squared

the pivotal columns in original matrix

x bar parallel is

the orthogonal projection of x bar onto the line

two vectors are orthogonal to each othe…

u * v = 0

Linear Equation

Linear System

What does RREF mean?

Pivot Position

An equation that can be written in the form a₁x₁+a₂x₂+...+aⁿxⁿ…

Collection of one or more linear equations

Row Reduced Echelon Form

A location in a matrix A that corresponds to a leading one in…

Linear Equation

An equation that can be written in the form a₁x₁+a₂x₂+...+aⁿxⁿ…

Linear System

Collection of one or more linear equations

Scalar multiplication

Standard Basis in Rn

Closed under addition

Closed under scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t(…

Denoted by

**e**n, [1, 0, 0, 0...0] to the nth number. So in R3,…Two vectors, when added together, are still in the same subspa…

When a vector is multiplied by a scalar, it's still in the sam…

Scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t(…

Standard Basis in Rn

Denoted by

**e**n, [1, 0, 0, 0...0] to the nth number. So in R3,…Trivial Solution

Nontrivial Solution

Consistent System

Homogeneous System

The zero vector

An answer besides the zero vector. Exists if and only if there…

There is at least one solution to the system

System in which all the numbers on the right hand side are zer…

Trivial Solution

The zero vector

Nontrivial Solution

An answer besides the zero vector. Exists if and only if there…

Consistent System

Inconsistent System

Properties of Matrix Addition and Scala…

Properties of Zero Matricies

A system of equations that has at least one solution

A system of equations that has no solution

1. A+B = B+A... 2. A+(B+C) = (A+B)+C... 3. (cd)A = c(dA)... 4. IA = A... 5…

1. A+0 = A... 2. A+(-A) = 0... 3. If cA = 0, then c=0 or A=0

Consistent System

A system of equations that has at least one solution

Inconsistent System

A system of equations that has no solution

equivalent system of equations

A Homogeneous system

invertible matrix

singularity

when two system of equations have the same solution set

when the constants on the right hand side of the matrix are al…

when there exists a matrix B such that AB=BA=I.

when a nxn matrix does not have a multiplicative inverse.

equivalent system of equations

when two system of equations have the same solution set

A Homogeneous system

when the constants on the right hand side of the matrix are al…

Consistent System

Inconsistent System

Consistent Dependent System

Free variable

The matrix system has a solution

Matrix has no solution (last row 0 0 | non zero )

Infinite systems (last row 0 0 | 0 )

x1 x2 x3 etc can be any number, no set equation, no pivot in t…

Consistent System

The matrix system has a solution

Inconsistent System

Matrix has no solution (last row 0 0 | non zero )

The number of solutions in a system of…

A system is said to be CONSISTENT if...

A system is said to be INCONSISTENT if.…

Echelon Form

None, One, or Infinitely Many

it has one or infinitely many solutions

it has no solution

1- rows of zeroes on the bottom, 2- each leading entry of a ro…

The number of solutions in a system of…

None, One, or Infinitely Many

A system is said to be CONSISTENT if...

it has one or infinitely many solutions

Linear combination

Linear Equation

System of linear equations

Solution

an expression of the form c[1]A[1]+c[2]A[2]+...+c[n]A[n] where…

a linear combination of unknowns (or variables) set equal to a…

a finite set of linear equations that all use the same variabl…

an assignment of values to the variables that turns all of the…

Linear combination

an expression of the form c[1]A[1]+c[2]A[2]+...+c[n]A[n] where…

Linear Equation

a linear combination of unknowns (or variables) set equal to a…

Solution Set

Consistent System

Inconsistent System

Elementary Row Operations

What x and y equal after matrix has been reduced and solved

Solution Set contains a single point or infinitely many points…

Solution Set is empty

Replacement: Replace a row with the sum of itself and a multip…

Solution Set

What x and y equal after matrix has been reduced and solved

Consistent System

Solution Set contains a single point or infinitely many points…

Hilbert-Schmidt norm (for a matrix) (ak…

Matrix (operator) norm

Vandermonde matrix

Formula for the determinant of a Vander…

concatenate all the columns of the matrix end-to-end and take…

A measure of the maximum amount by which a matrix "stretches"…

Matrix used in Lagrangian interpolation (in 2D). Each row corr…

Plus or minus the product of the pairwise differences of every…

Hilbert-Schmidt norm (for a matrix) (ak…

concatenate all the columns of the matrix end-to-end and take…

Matrix (operator) norm

A measure of the maximum amount by which a matrix "stretches"…

linear equation

system of linear equations (linear syst…

solution

A system of linear equations has

an equation that can be written in the form ax+ax+ax+...+ax=b…

a collection of one or more linear equations involving the sam…

A solution of the system is a list (s,s,s,s..,s) of numbers th…

1. no solution... 2. exactly one solution... 3. infinitely many solu…

linear equation

an equation that can be written in the form ax+ax+ax+...+ax=b…

system of linear equations (linear syst…

a collection of one or more linear equations involving the sam…

Thm. 4.2.1 - If B is produced from A by…

Thm. 4.2.1 - If B is produced from A by…

Thm. 4.2.1 - If B is produced from A by…

Thm. 4.2.2 - If A has two identical row…

det(B) = kdet(A)

det(B) = -det(A)

det(B) = det(A)

det(A) = 0

Thm. 4.2.1 - If B is produced from A by…

det(B) = kdet(A)

Thm. 4.2.1 - If B is produced from A by…

det(B) = -det(A)

(VS1) ... Commutativity of Addition

(VS2)... Associativity of Addition

(VS3)... Zero Vector;... Identity Element of…

(VS4)... Inverse Elements of Addition

For all x, y in V, x+y = y+x

For all x, y, z in V (x+y)+z = x+ (y+z)

There exists an element in V denoted by 0 such that x+0=x for…

For each element x in V there exists and element y in V such t…

(VS1) ... Commutativity of Addition

For all x, y in V, x+y = y+x

(VS2)... Associativity of Addition

For all x, y, z in V (x+y)+z = x+ (y+z)

Consistent System

Inconsistent System

Row Equivalent Matrices

Linear Combination

The system has either 1 solution or infinitely many

The system has no solutions

If a sequence of Elementary Row Operations can transform one i…

Given vectors in R^n and scalars c1....cp, the vector b is giv…

Consistent System

The system has either 1 solution or infinitely many

Inconsistent System

The system has no solutions

vector space X over 𝐹

subspace

linear combination

span of a subset of vectors .

is a nonempty set with two operations, "addition" and "scalar…

Vector spaces may be formed from subsets of other vectors spac…

one vector is equal to the sum of scalar multiples of other ve…

is the set of all possible linear combinations of u1,u2,u3,...…

vector space X over 𝐹

is a nonempty set with two operations, "addition" and "scalar…

subspace

Vector spaces may be formed from subsets of other vectors spac…

Theorem 4.7.2

Theorem 4.7.3

Theorem 4.7.4

Theorem 4.7.5

If x0 is any solution of a consistent linear system Ax=b, and…

Elementary row operation do not change the null space of a mat…

Elementary row operations do not change the row space of a mat…

if a matrix R is in row echelon form, then the row vectors wit…

Theorem 4.7.2

If x0 is any solution of a consistent linear system Ax=b, and…

Theorem 4.7.3

Elementary row operation do not change the null space of a mat…

linear equation

slope

Types of Slope

y-intercept

An equation in two variables whose graph in a coordinate plane…

A ratio comparing the change in output over the change in inpu…

The value of the output when the input is zero (0, y). The poi…

linear equation

An equation in two variables whose graph in a coordinate plane…

slope

A ratio comparing the change in output over the change in inpu…

Solving if something is in a Span

Span Theorems

Span

Subset

By definition of span, we want to know if there are scalars x1…

Theorem 1: The subspace spanned by a non-empty subset S of a v…

The set of all linear combinations of the vectors.... or ... The s…

A subset is a set of vectors. Assume a subset V∈ℜn... Can be a s…

Solving if something is in a Span

By definition of span, we want to know if there are scalars x1…

Span Theorems

Theorem 1: The subspace spanned by a non-empty subset S of a v…