Matrices and Linear Transformations Quiz 1 Terms and Definitions

complex number

a+bi where a,b∈R and i² = −1.

C

The notation for the set of all complex numbers.

complex number addition

basically vector addition in complex plane

complex number multiplication (rectangular)

Foil as you would do with binomials

complex number multiplication (polar)

magnitudes are multiplied and angles are added
zw=rs(cos(θ+Φ)+isin(θ+Φ))

complex conjugate (a + bi)

a - bi

The magnitude of a complex number (a + bi)

Distance from 0 is the complex plane calculated using the formula ( |z| = \sqrt{a^2 + b^2} ).

rectangular form of a complex number

a + bi

polar form of a complex number

r(cosθ+isinθ)

exponential form of a complex number

re^iθ based on cosθ+isinθ

nth root of unity

The solutions to the equation zⁿ=1

vector field

A set that is closed under addition and multiplication and satisfies:
Addition: commutative (A1), associative (A2), has an identity element 0 (A3), and every element has an inverse (A4).
Multiplication: commutative (M1), associative (M2), has an identity element 1 (M3), and every non-zero element has an inverse (M4).

Distributive Law: x(y+z)=xy+xz (D).

associative

For addition in a field or vector space:

(x+y)+z=x+(y+z).
For multiplication in a field:

(xy)z=x(yz)

commutative

For addition in a field or vector space:

x+y=y+x
For multiplication in a field:

xy=yx

distributive

In a field:

x(y+z)=xy+xz

additive inverse

In a field: For every element x, there exists an element −x such that x+(−x)=0
In a vector space: For each vector v, there is a vector −v such that v+(−v)=0.

multiplicative inverse

For every non-zero element x, there exists an element x^-1 such that x⋅x^-1 = 1.

additive identity

In a field: An element 0 such that for all x, x+0=x.
In a vector space: A "zero vector" 0 such that for all v, v+0=v.

multiplicative identity

In a field: An element 1 (where 1=0) such that for all x, 1⋅x=x
In a vector space: The scalar 1 from the associated field has the property that 1v=v for any vector v.

vector space

V with an associated field of scalars F that is closed under vector addition and scalar multiplication and satisfies 8 axioms:

1. Addition is commutative (VS1).

2. Addition is associative (VS2).

3. There is an additive identity (zero vector) (VS3).

4. Every vector has an additive inverse (VS4).

5. There is a multiplicative identity for scalar multiplication (VS5).

6. Scalar multiplication is "associative" (VS6).

7. Scalar multiplication distributes over vector addition (VS7).

8. Scalar addition distributes over scalar multiplication (VS8).

linear combination

α1​v1​+α2​v2​+...+αm​vm​ where v1​,...,vm​ are vectors and α1​,...,αm​ are scalars.

basis

A set of vectors

{v1​,v2​,...,vn​} is a basis for a vector space V if every vector v∈V can be written as a linear combination of these vectors in exactly one way.
needs to be linearly independant

coordinates (with respect to a basis)

The scalar coefficients

α1​,...,αn​ in the unique representation of a vector as a linear combination of basis vectors, i.e., v=α1​v1​+...+αn​vn​.

span

The span of a set of vectors

{v1​,...,vn​} is the set of all possible linear combinations of those vectors.

generating (or spanning or complete) set

A set of vectors

{v1​,v2​,...,vn​} is called a generating set for a vector space V if every vector in V can be written as a linear combination of them. This means that the span of the set is equal to the entire vector space V.

trivial linear combination

A linear combination where all the scalar coefficients are zero.

linearly dependent

if vectors in the set can be written as a linear combination of the others

linearly independent

if no vector in the set can be written as a linear combination of the others