# Study sets matching "linear algebra"

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,…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…

Suppose we have a final solution set of…

How are two linear systems equivalent?

How do you solve a linear system?

A matrix with size 3x4 is: ____________…

infinite

They share the same solution sets

Replace the l.s. with one that is equivalent but "simpler", th…

augumented matrix of *

Suppose we have a final solution set of…

infinite

How are two linear systems equivalent?

They share the same solution sets

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…

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,…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…

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

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

3.1 Two equivalent vectors must have th…

3.1 The vectors (a,b) and (a,b,0) are e…

3.1 If k is a scalar and v is a vector,…

3.1 The vectors v+(u+w) and (w+v)+u are…

F (옮겨서 시점이 달라져도 같은 벡터로 취급한다.)

F (일단 기본적으로 차원이 같아야 함.)

F (벡터가 평행하다는 것은 굳이 방향이 같을 필요는 없음. 방향은 반대더라도 됨.)

T

3.1 Two equivalent vectors must have th…

F (옮겨서 시점이 달라져도 같은 벡터로 취급한다.)

3.1 The vectors (a,b) and (a,b,0) are e…

F (일단 기본적으로 차원이 같아야 함.)

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…

If dot product of u and v = 0, ( u*v =…

the norm of a vector is ??

the distance between two vectors is ??

what is || v || and what does it equal…

perpendicular or orthogonal

the length of the vector

||v-u||

the norm and is equal to sqrt(v^2+v^2+v^2...)

If dot product of u and v = 0, ( u*v =…

perpendicular or orthogonal

the norm of a vector is ??

the length of the vector

Matix

Elements of the matrix

Submatrix

Square matrix

A rectangular array of numbers.

The numbers in the array

A matrix composed of a array of numbers taken from a larger ma…

When there are equal numbers of rows as Columns

Matix

A rectangular array of numbers.

Elements of the matrix

The numbers in the array

Linear combination

Linearly independent

Theorem 5.1.1

elementary vectors

an expression of the form c1v1+c2v2+...+cnvn where all the ci…

a set of vectors in LI if the only way to write the zero vecto…

A set of vectors is LD iff one of the vectors can be written a…

vectors that have exactly one component equal to 1 and all oth…

Linear combination

an expression of the form c1v1+c2v2+...+cnvn where all the ci…

Linearly independent

a set of vectors in LI if the only way to write the zero vecto…

Characterization of Linearly Dependent…

Multiplication Property

An inverse formula

Existence and Uniqueness theorem

An indexed set S={V1,.....Vp} of 2 or more vectors is linearly…

If A and B are nxn matrices then detAB=detA*detB

Let A be an invertible nxn matrix then A^-1 = (1/detA)adjA

A linear system is consistent if and only if the rightmost col…

Characterization of Linearly Dependent…

An indexed set S={V1,.....Vp} of 2 or more vectors is linearly…

Multiplication Property

If A and B are nxn matrices then detAB=detA*detB

linear equation

system of linear equations

solution set

equivalent linear systems

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

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

a list of numbers that makes each equation in a system a true…

linear systems with the same solution set

linear equation

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

system of linear equations

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

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…

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.

A finite basis for V of Field F is a fi…

Let S be a nonempty subset of a vector…

Let V & W be vector spaces over field F…

A nonempty finite subset {u1, u2,.....,…

Finite Basis

Span

Linear Transformations (Operators)

Linear Dependent

A finite basis for V of Field F is a fi…

Finite Basis

Let S be a nonempty subset of a vector…

Span

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.

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,…Elementary Matrix

property of Elementary Matrix

Cramer's Rule2

subspace of Rn

what an n x n matrix E is called if we can obtain E from I, by…

each of these elementary row operations can be implemented by…

(a) the sum of any pair of vectors in the set lies in the set,…

Elementary Matrix

what an n x n matrix E is called if we can obtain E from I, by…

property of Elementary Matrix

each of these elementary row operations can be implemented by…

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,…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.