Linear Algebra II Exam 1

T/F: Lists have a finite length

True

subspace conditions

0 is in U, vector addition, scalar multiplication

sums of subsets

the set of all possible sums of elements of the subsets (the smallest containing subset)

direct sum

if U intersecting W = {0}

degree of p = 0

negative infinity

span of the empty list () =

{0}

linear map (T: V→ W)

the following properties hold:

T(u+v) = T(u) + T(v)

T(cv) = cT(v)

null space

subset of V consisting of the vectors that T maps to 0, subspace of V

injective

if Tu = Tv implies u = v (one-to-one)

nullT = {0}

range

image, subspace of W

surjective

if range = W, onto

Fundamental Theorem of Linear Maps

dimV = dim(nullT) + dim(rangeT)

map onto a smaller dimensional space

not injective

map onto a larger dimensional space

not surjective

dim (F^m,n)

m\*n

matrix multiplication properties

associative, distributive

invertible

if matrix times inverse is identity and commutative is also true

is the inverse unique?

yes

isomorphism

invertible linear map, domain space is the same dimension as range space

isomorphic

if there’s an isomorphism from a vector space onto the other one

two lists are equal iff…

they have the same length and the same elements in the same order

span of a list of vectors

the smallest subspace of V containing all the vectors in the list

linearly independent

if the only choice of a1 - am, to make a1v1 + … + amvm = 0, is a1 = am = 0

linear dependence lemma

if v1, …, vm is linearly dependent, then there exists vj in span(v1, …, vj-1) and the span of the original list stays the same with vj removed

length of linearly independent list is

less than or equal to the length of spanning list

basis

list of vectors that are linearly independent and span V

every subspace of V is part of

a direct sum equal to V

every linearly independent list of vectors in V with length (dimV) is

a basis of V

dim(U1 + U2) =

dimU1 + dimU2 - dim (intersection of U1+2)

there exists a unique linear map from V to W that gets from the basis of V vectors

to the vectors in the basis of W

if T: V→W, what space is nullT a subspace of?

V

homogeneous

right side of each equation is 0

invertibility is equivalent to

injectivity and surjectivity (ie, only invertible if injective and surjective)

dim (L(V,W) =

dimV \* dimW

operator

linear map from a vector space to itself

product of vector spaces

{(v1, …, vm) | v1 is in V1, …, vm is in Vm}, also a vector space

dimension of a product of vector spaces is

the sum of their dimensions

sum is a direct sum iff

dim(U1 + … + Um) = dim(U1) + … + dim(Um)

v + U (U is a subspace of V)

= {v+u | u is in U}, subset of V

affine subset

subset of V of the form v+U

parallel

v+U is parallel to U

quotient space (V/U)

{v + U | v in V}

(v+U) + (w+U)

(v+w)+U

c(v+U)

cv + U

quotient map denoted by pi

pi: V → V/U

pi(v) = v + U

dimension of V/U

dimV - dimU

T\~

V/nullT →W

T\~(v+nullT) = Tv

T\~ properties

injective, rangeT\~ = rangeT, domain is isomorphic to rangeT