# 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

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,…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

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)

what is a linear combination?

what is the span of a vector set?

What is a basis?

what is a translation from one coordina…

It's a vector that can be reached by adding and scaling other…

It's the set of all linear combinations of a group of vectors…

A basis is a set of vectors belonging to some space ... *Intuiti…

Intuitively it's transforming to a different coordinate system…

what is a linear combination?

It's a vector that can be reached by adding and scaling other…

what is the span of a vector set?

It's the set of all linear combinations of a group of vectors…

What is a subspace and what are the req…

Column Space?

What is closure?

What is true for the span of a set of v…

A subset (H) of a larger set of vectors(V) that has 3 properti…

denoted col(A)... the span of the columns of A.

When an operation such as addition is done with two numbers fr…

For a set of vectors {v₁, v₂, v₃....} in Rⁿ... span{v₁,v₂,v₃...}…

What is a subspace and what are the req…

A subset (H) of a larger set of vectors(V) that has 3 properti…

Column Space?

denoted col(A)... the span of the columns of A.

Reduced row echelon form conditions

Theorem "dimension characterizes isomor…

Define "W is isomorphic to V"

Theorem stating "isomorphic to" is an e…

1. leading entry (if there is one) is 1 2. if column contains…

For subspaces V within R^n and W within R^m, we have V is isom…

There exists an isomorphism T:W->V

"isomorphic to" is an equivalence relation for subspaces V wit…

Reduced row echelon form conditions

1. leading entry (if there is one) is 1 2. if column contains…

Theorem "dimension characterizes isomor…

For subspaces V within R^n and W within R^m, we have V is isom…

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…

Algebraically, the value of the output when the input is zero…

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…

Free Variable Theorem for Homogeneous s…

a homo geneous linear system with more…

If B and C are both inverses of the mat…

If A and B are invertible matrices with…

If a homogeneous linear system has n unknowns, and if the redu…

infinitley many solutions

since B is an inverse of A, we have BA=I. Multiplying both sid…

(AB)(B^-1A^-1)=(B^-1A^-1)(AB)=I , but ... (AB)(B^-1A^-1)=A(BB^-1)…

Free Variable Theorem for Homogeneous s…

If a homogeneous linear system has n unknowns, and if the redu…

a homo geneous linear system with more…

infinitley many solutions

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…

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"…

Rank

The Rank Theorem

Dimension Col(A)

Dimension Row(A)

The __________ of A is the dimension of the column space of A

Rank A + dim Nul(A) = n... {Number of pivot columns} + {Number of…

Number of pivot positions

Number of non-zero rows

Rank

The __________ of A is the dimension of the column space of A

The Rank Theorem

Rank A + dim Nul(A) = n... {Number of pivot columns} + {Number of…

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

Matrix Equation

Linear Combination

Span

Basis (2.8)

Ax = b

Scaling and adding a set of vectors.

The set of ALL possible linear combinations for a set of vecto…

Linearly independent vectors that are scaled to create the ent…

Matrix Equation

Ax = b

Linear Combination

Scaling and adding a set of vectors.

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…

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…

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 )