# Study sets matching "linear algebra"

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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,…Vector Space First Axiom

Vector Space Second Axiom

Vector Space Third Axiom

Vector Space Fourth Axiom

for all x,y under the set of V, x+y=y+x

for all x,y,z under the set of V, (x+y)+z=x+(y+z)

A zero vector exists in V, for all x in the set of V, x+(the z…

for all x under the set of V, the exists y under the set of V…

Vector Space First Axiom

for all x,y under the set of V, x+y=y+x

Vector Space Second Axiom

for all x,y,z under the set of V, (x+y)+z=x+(y+z)

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

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…

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…

There are only three types of solutions…

A system of linear equations is...

A system is called homogeneous if...

A system is called consistent if..

1. no solution. (parallel, no intersection, same slope)... 2. on…

is a list of linear equations in the same variables.

if each equation in the system is equal to zero. ... These system…

if there exists at least one solution. ... -a unique solution, wh…

There are only three types of solutions…

1. no solution. (parallel, no intersection, same slope)... 2. on…

A system of linear equations is...

is a list of linear equations in the same variables.

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.

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…

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…

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…

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,…Absolute Value

Integer

Exponent

Function

a number's distance from zero

positive and negative whole numbers

tells how many times to multiply the base number

a relationship where the input determines the output

Absolute Value

a number's distance from zero

Integer

positive and negative whole numbers

Vector Space

Vector Space Axioms

Subspace

Linear Combination

A nonempty set V of vectors following 10 axioms for vectors u,…

1: u + v is in V... 2: u + v = v + u... 3: (u + v) + w = u + (v + w)…

A subspace H of vector space V is a subset that has three prop…

Any sum of scalar multiples of vectors

Vector Space

A nonempty set V of vectors following 10 axioms for vectors u,…

Vector Space Axioms

1: u + v is in V... 2: u + v = v + u... 3: (u + v) + w = u + (v + w)…

linear equation

slope

y-intercept

x-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…

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

coordinate plane

origin

x-axis

y-axis

the grid created by the horizontal x-axis and the vertical y-a…

(0,0) in the coordinate plane; center of the grid

the horizontal number line in the coordinate plane (left to ri…

the vertical number line in the coordinate plane (up and down)

coordinate plane

the grid created by the horizontal x-axis and the vertical y-a…

origin

(0,0) in the coordinate plane; center of the grid

If the augmented matrices of 2 linear s…

Another way to ask if a system is consi…

Echelon Form (properties)

Reduced Echelon Form (properties)

the 2 have the same solution set

does at least 1 solution exist

All non zero rows above zero rows... each leading entry of a row…

All properties from Echelon Form... leading entry is 1 in each n…

If the augmented matrices of 2 linear s…

the 2 have the same solution set

Another way to ask if a system is consi…

does at least 1 solution exist

In the equation, y = mx + b, the m stan…

In the equation, y = mx + b, the b stan…

In the equation y = 3x + 7, the slope i…

In the equation, y = −2x − 6, the y int…

Slope

y-intercept

3

−6

In the equation, y = mx + b, the m stan…

Slope

In the equation, y = mx + b, the b stan…

y-intercept