Linear Algebra Definitions Semester 1

Group

Set of elements under 1 operation such that there is identity, inverse, closure, and associativity

Field

Group under two operations, usually either the reals or complex numbers. Elements are used as scalars. Symbol= F. R and C are fields

C

all complex numbers

F^n

Set of all lists of length n of elements of R or C. Eg: C^4 is the set of all lists of four complex numbers.

Vector Space

Set V along with an addition and scalar multiplication with:

Commutativity, (u+v = v+u) for u, v ∈ V

Associativity, ((u + v)=w = u + (v + w) and (ab)v = a(bv) for all u, v, w ∈ V and a, b ∈ F

Additive identity, (v + 0= 0)

Additive inverse, (some w ∈ V so v + w = 0)

Multiplicative identity, (1v = v)

Distributive properties, a(u + v) = au + av and (a + b)v = av +bv.

Subspace

A subset U of V is called a subspace of V if U is also a vector space (using the same addition and scalar multiplication as on V).

Subspace Criteria

Additive identity

Closed under addition (u, w ∈ U means u+w ∈ U)

Closed under scalar multiplication (a ∈ f and u ∈ U means au ∈ U)

Sum of subsets

The sum of U1,....Um is the set of all possible sums of elements of U1,....,Um.

Direct Sum

If V is the direct sum of U and W, then each vector in V can be expressed as a unique sum of a vector from U and V.

Condition for a direct sum

if U1,...,Um are subspaces of V, then U1 +...+ Um is a direct sum if and only if the only way to get 0 from the sum um +..+ um is by taking each uj = 0.

Direct sum of two subspaces in V

can only occur if U ∩ W = {0}

Linear Combination

Adding scalar multiples of a list of vectors:
a linear combination of list v1,...,vm of vectors in V is a vector of the form a1v1+...+amvm
span(v1,...,vm)={a1v1+...+amvm : a1,...,am in F}

Span (noun)

Set of all linear combinations of a list of vectors. Span of empty list is {0}, it's also the smallest subspace of a vector space containing all vectors in the list.

Span (verb)

if the span of a list of vectors equals V, then that list spans V.

Polynomial

P(F) is set of all polynomials with coefficients in F

Pm(F)

For m a nonnegative integer, Pm(F) denotes the set of all polynomials with coefficients in F at degree most m

linearly independent

Only choice that makes a linear combination = 0 is if all coefficients = 0, aka you cant express one with the others

linearly dependant

Not linearly independent, can express 0 from a linear combination where not all coefficients are 0.

Linear Dependence Lemma

If there is a linearly dependent list, then there is some vj ∈ span(v1,......,vj-1). If jth term is removed from the list, the span stays the same.

Basis

Linearly independent spanning list. Every finite dimensional vector space has one.

any two bases of a finite-dimensional vector space have the same length

Dimension

Length of basis

Linear Map

A linear map from V to W is a function T : V-->W with:

Additivity:
T(u + v) =. Tu + Tv
Homogeneity:
T(av) = a(Tv)

L(V, W)

Set of all linear maps from V to W

addition and scalar multiplication on L(V,W)

(S + T)(v) = Sv + Tv and (aT)(v) = a(Tv)

Product of Linear Maps

(ST)(u) = S(Tu)

Null Space

Null space of T is the subset of V (domain) consisting of vectors that T maps to 0.

The Null space is a subspace

Injective

A function T: V--->W is injective if Tu = Tv implies u = v

Range

Range of T is the subset of W consisting of vectors that are of the form Tv for some v ∈ V. AKA all the vectors that T hits and doesn't go to the null space

Is a subspace of W

Surjective

A function T: V--->W is surjective if its range = W

Fundamental Theorem of Linear Maps

If V is finite dimensional and T ∈ L(V,W), then range T is finite dimensional and dim V = dim null T + dim range T

Relate bases, spanning sets, and linearly independent sets

Spanning set and linear independence are criteria for bases. A spanning set can also be reduced to a basis and a linearly independent set can be extended to one

Operator

Linear map from a vector space to itself, aka the same dimension.

Identity transformation

Transformation that maps every vector in v to itself: I(v) =. v

elementary matrices

An invertible matrix that results by performing one elementary row operation on an identity matrix

nonsingular matrix

a matrix that is invertible

Understand matrices as linear transformations

Every linear transformation is associated with a matrix

geometric interpretation of dot product

Based on projection of vector onto another. Dot product divided by magnitude = projection length, this in turn = magnitude of other vector times cos of angle between them. So dot product = magnitude of each vector multiplied by cos of angle between them. Shows how much one vector is pointing in the direction of another.

Find an inverse matrix from elementary matrices

apply the fundamental theorem of linear maps

Can be used to prove that a map to a smaller dimensional space is not injective and a map to a larger dimensional space is not surjective.

apply concepts of span and basis

Both span and basis are used to express a vector space, but a span can be linearly dependent.

Linearity

For a system to be linear, it must be both homogeneous and additive.

Elementary row operations

Row Swap) Exchange any two rows. (Scalar Multiplication) Multiply any row by a constant. (Row Sum) Add a multiple of one row to another row.

geometric interpretation of cross product

vector orthogonal to a and b. Magnitude= the area of the parallelogram they span (aka their distance).

geometric interpretation of the determinant

shows how a linear transformation associated with a matrix can reflect or scale an area.

Vector space is over something (x)

use x as coefficients in linear combinations

Invertible

A linear map T ∈ L(V,W) is invertible if there exists a linear map S ∈ L(W,V) such that ST equals the identity map on V and TS equals the identity map on W.

Linear independence and system of equations

equations only have one solution. If linearly dependent, have infinite solutions because they refer to the same line.

Linear independence and direct sum

in a direct sum, non-zero vectors taken from the different subspaces being summed must be linearly independent.

Isomorphism

An invertible linear map

Linear maps and basis of domain

If v1,....vn is a basis of V and w1,....wn ∈ W. Then there exists a unique linear map T:V ---> W such that Tvj = wj, AKA, once you know what a transformation does to a basis, you know what it does to the rest of the space.

Isomorphic

Two vector spaces are called isomorphic if there is an isomorphism between them

Inverse of Matrix

only exists if determinant doesnt equal 0.

Matrix of linear map

Matrix of linear map T is m by n matrix where entries are defined by Tvk = A1,k w1+.....+Am, kWm where w1,,,wm is the basis of W and v1,...vn is the basis of V

Compute Cross Product

a × b = |a| |b| sin(θ) n OR cx = aybz − azby; cy = azbx − axbz, cz = axby − aybx

Compute Dot Product

a · b = |a| × |b| × cos(θ); (abc) · (def) = (a · d) + (b · e) + (c · f)

Properties of complex arithmetic

Commutativity, Associativity, Identities, Inverses, Distributive property

list

a list of length n is an ordered collection of n elements

Criterion for a Basis

A list v1...vn of vectors in V is a basis of V if and only if every v in V can be written uniquely in the form v=a1v1+anvn

Spanning list contains a ?

basis

A linear Independent list expands to a ?

basis (and spanning list)

If V is finite dimensional and U is a subspace of V what is the relationship between their dimensions?

dim U is less than or equal to dim V

Linear Independent List is right length

it is basis

Spanning List is right length

it is basis

If U1 and U2 are subspaces of a finite dimensional vector space then

dim(U1+U2)=dimU1+dimU2-dim(U1intersection with U2)