Linear maps

Question 1

linear map

Answer 1

if V and W are 2 vector spaces then a linear map T:V-> W is a function satisfying:
- T(v1+v2) = Tv1 + Tv2
- T(µv) = µT(v)

Question 2

how to determine a linear map using basis vectors

Answer 2

let {v1…vn} be basis for V then:
any linear map T is determined by the values T(vi)
given n elements w1..wn in W there is exactly one linear map with T(vi) = wi

Question 3

linear map and matrix

Answer 3

a function T is a linear map from R^n to R^m then there is an mxn matrix A such that
T(v) = Av
A = (T(e1)…T(en))

Question 4

composition of linear maps

Answer 4

if T is V-> W and S is W->U then ST: V->U is also a linear map
also if T and S are represented by A and B matrixes then ST is represented by BA

Question 5

1-1

Answer 5

if f(x1) = f(x2) then x1 = x2
cant have two elements of x map by f to the same y

Question 6

onto

Answer 6

there is an x in X with f(x) = y

Question 7

bijection

Answer 7

function that is 1-1 and onto

Question 8

linear map as bijection

Answer 8

if T is a linear map and a bijection of vector spaces then its inverse is also a linear map
if from R^n - R^n then and represented by A then inverse represented by A^-1

Question 9

isomorphism

Answer 9

bijective linear map

Question 10

coordinate map as an isomorphism

Answer 10

if V is an n dimensional vector space
chose an ordered basis B then the coordinate map is an isomorphism
in particular any abstract vector space with a spanning set is isomorphic to R^n for some n

Question 11

square diagram theorem

Answer 11

let V be vector space dimension n and W dimension m. suppose we have chosen ordered bases B,C of V and W giving coordinate maps øb and øc
suppose T a linear map V-> W there is a matrix A that represents it
V->W is T
V-> R^n is øb
W-> R^m is øc
R^n -> R^m is A
øcT(v) = A(øb v)

Question 12

ith column of a matrix representing a linear map

Answer 12

give a map µfrom R^n to R^m the matrix that represents it has ith colum µ(ei)
the matrix A has ith column ø(T(vi))

Question 13

image

Answer 13

ImT = {w in W | there is a v in V with T(v) = w}

Question 14

kernel

Answer 14

kerT = {v in V|Tv = 0}

Question 15

image and kernel as vector subspaces

Answer 15

is T: V-> W then ImT is a vector subspace of W and kerT is a vector subspace of V

Question 16

spanning set for image

Answer 16

T: V-> W is a linear map and {v1…vn} is a basis for V
then {Tv1…Tvn} is a spanning set for ImT

Question 17

equivalent statements to ketT = {0}

Answer 17

let T:V-> W be a linear map then the following are equivalent
- kerT = {0}
- T is 1:1 (ie Tv1 = Tv2 then v1 = v2)
- if {v1..vm} is an LI set in V then {Tv1…Tvm} is an LI set in W

Question 18

basis for image

Answer 18

let T : V-> W be a linear map and kerT = {0}
if {v1…vn} basis for V then {Tv1…Tv2} basis for ImT

Question 19

rank of a linear map

Answer 19

rkT is the dim(ImT)

Question 20

nullity of a linear map

Answer 20

nullT is the dim(kerT)

Question 21

composite of isomorphisms

Answer 21

let T be a linear map V->W
suppose we have isomorphism V’->V by ø and W->W’ by ψ
can form the composite
V’->V->W->W’ by ø then T then ψ
then rank(ψTø) = rank(T)
null(ψTø) = null(T)

Question 22

rank nullity theorem

Answer 22

let V and W be finite dimensional vector spaces and T: V->W a linear map then
dimV = rankT + nullT

Question 23

isomorphism and dimensions

Answer 23

there is an isomorphism from V to W implies dimV = dimW