# 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,…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

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…

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

System of Linear Equations

Solution

Solution Set

describes a firsrt order line in Rn equations of the form a1x1…

collection of 1+ linear equations to be solved simultaneously

a set of real numbers s1...sn such that each equation holds if…

a set of all solutions in a system

Linear Equation

describes a firsrt order line in Rn equations of the form a1x1…

System of Linear Equations

collection of 1+ linear equations to be solved simultaneously

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,…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

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…

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…

Geometric Vector.

Why are physical vectors defined unique…

Physical Vector.

The Parallelogram Law for Addition of G…

An arrow in space centered at the origin whose length indicate…

Dir and mag are usually the only important features in physica…

Entities that have a magnitude and direction in physical life,…

The sum of the vectors x and y that act on a point P is the ve…

Geometric Vector.

An arrow in space centered at the origin whose length indicate…

Why are physical vectors defined unique…

Dir and mag are usually the only important features in physica…

Linear Equation

System of Linear Equations

Solution of System

Linear Systems

a1x1+a2x2+...+anxn=b ... b and coefficients are real numbers

collection of one or more linear equations with the same varia…

Ordered list of numbers which make each of the equations true…

can be: ... -no sol'n (inconsistent)... -1 sol'n (consistent, unique…

Linear Equation

a1x1+a2x2+...+anxn=b ... b and coefficients are real numbers

System of Linear Equations

collection of one or more linear equations with the same varia…

Rate of change

Slope

Direct variation

Constant of variation for a direct vari…

The relationship between two quantities that are changing. The…

The ratio of the vertical change to the horizontal change.... slo…

A linear function defined by an equation of the form... y = kx, w…

The nonzero constant k in the function y = kx.

Rate of change

The relationship between two quantities that are changing. The…

Slope

The ratio of the vertical change to the horizontal change.... slo…

associative property of addition

coefficient

commutative property of addition

constant

(a + b) + c = a + (b + c)

the number that is multiplied by the variable

a + b = b + a

a term that has no variable, a single number

associative property of addition

(a + b) + c = a + (b + c)

coefficient

the number that is multiplied by the variable

Homogeneous Linear System

Trivial/Nontrivial Solutions

The homogeneous equation Ax=0 has a non…

A nontrivial solution x can have some z…

1. A system of linear equations is said to be homogeneous if i…

1. The zero solution is called the trivial solution... 2. The non…

the equation has at least one free variable.

not all of the entries are zero.

Homogeneous Linear System

1. A system of linear equations is said to be homogeneous if i…

Trivial/Nontrivial Solutions

1. The zero solution is called the trivial solution... 2. The non…

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)

homogeneous linear equation

trivial solution to homogeneous

nontrivial solution to homogeneous

if zero vector is only solution

Ax = 0

vector x = 0... Span{0}

nonzero vector x that satisfies Ax = 0... can only exist IFF the…

solution set is Span{0}

homogeneous linear equation

Ax = 0

trivial solution to homogeneous

vector x = 0... Span{0}

Order of Operations

Inequality

Like Terms

Equivalent expressions

Ch. 1 PEMDAS the order in which you solve an equation. (Parent…

Ch. 1 An open sentence that contains the symbols < >

Ch. 1 Terms that contain the same variables, with correspondin…

Ch. 1 Expressions that simplify to the same number.

Order of Operations

Ch. 1 PEMDAS the order in which you solve an equation. (Parent…

Inequality

Ch. 1 An open sentence that contains the symbols < >