Exam 2024 (B)

1a

Let V,W be finitely generated vector spaces over FLet T:V→W be a linear transformationProve: T is invertible ⟺T is bijectiveProof:Let T be invertible⟹∃linear transformation T−1:W→V,TT−1=IW,T−1T=IVTT−1=IW⟹TT−1 is surjective⟹T is surjectiveT−1T=IV⟹T−1T is injective⟹T is injective⟹T is bijectiveLet T be bijectiveT is a function⟹∃ function S:V→W,TS=IW,ST=IVLet w1,w2∈W,α∈FS(w1+αw2)=ST(S(w1+αw2))=S(TS(w1+αw2))=S(TS(w1)+αTS(w2))==S(T(S(w1)+αS(w2)))=ST(S(w1)+αS(w2))=S(w1)+αS(w2)⟹S is a linear transformation⟹T is invertible

1b

Let T:R2→R2T((xy))=(2xx+y)Let B={(11),(21)} basis of R2Find [T]BBSolution:[T]BB=([T(11)]B[T(21)]B)=([(22)]B[(43)]B)=(2201)

2

U=sp{(0001),(1101)}W={(abcd)|a−d=0a+b+c−2d=0}Find basis and dimension of U+W,U∩WSolution:W={(abcd)|a−d=0a+b+c−2d=0}={(dbcd)|b+c=d}={(b+cbcb+c)}==sp{(1101),(1011)}⟹U+W=sp{U∪W}=sp{(0001),(1101),(1011)}{(0001),(1101),(1011)} is a linear independence and is a basis of U+W⟹dim(U+W)=3Let v∈U∩W⟹α(0001)+β(1101)=γ(1101)+δ(1011)⟹(β−γ−δβ−γδα+β−γ−δ)=(0000)⟹δ=0⟹β=γ⟹α=0⟹v=(ββ0β)⟹U∩W=sp{(1101)}⟹{(1101)} is a basis of U∩W and dim(U∩W)=1Alternative solution:sp{(0111)}⊆U and sp{(0111)}⊆W⟹sp{(0111)}⊆U∩Wdim(U+W)⏟3=dim(U)⏟2+dim(W)⏟2−dim(U∩W)⟹dim(U∩W)=1⟹sp{(0111)}=U∩W

3a

Let T:R3→R3 be a linear transformationT(10−1)=(121),T(123)=(111),T(222)=(232)How many linear transformations T satisfy these conditions?Solution:(222)=(10−1)+(123)T(222)=T(10−1)+T(123)⟹Such T existsLet B={(10−1),(123),v3} be a basis of R3⟹By the defining theorem ∀u∈R3:∃!T:{T(10−1)=(121)T(123)=(111)T(v3)=u⟹There exists an infinite number of such linear transformations

3b

Let V be a finitely generated vector spaceLet T,S:V→V be linear transformationsProve: dim(ker(TS))≤dim(ker(T))+dim(ker(S))Proof:Let R:ker(TS)→V,R(v)=S(v)v∈ker(ST)⟹{T(S(v))=T(0)=0v∈ker(S)T(S(v))=0⟹S(v)∈ker(T)v∉ker(S)⟹Im(R)=ker(T)v∈ker(R)⟹R(v)=S(v)=0⟹v∈ker(S)⟹ker(R)=ker(ST)∩ker(S)⟹dim(ker(ST))=dim(ker(R))+dim(Im(R))==dim(ker(S)∩ker(ST))+dim(ker(T))ker(S)∩ker(TS)⊆ker(S)⟹dim(ker(S)∩ker(TS))≤dim(ker(S))⟹dim(ker(TS))≤dim(ker(S))+dim(ker(T))

4

Let V be a finitely generated vector space, dim(V)≥2Let T,S:V→V be linear transformations

4a

Prove or disprove: TS is surjective⟹T is an isomorphismProof:Let TS be surjectiveTS is surjective⟹T is surjectivedim(V)=dim(V)⟹T is injective⟹T is bijective⟹T is invertible (an isomorphism)

4b

Prove or disprove: Im(T+S)⊆Im(T)⟹Im(S)⊆Im(T)Proof:Let Im(T+S)⊆Im(T)Let u∈Im(S)∃v∈V:S(v)=u(T+S)(v)=T(v)+S(v)∈Im(T)⟹∃w∈V:T(w)=T(v)+S(v)⟹u=S(v)=T(w−v)⟹u∈Im(T)⟹Im(S)⊆Im(T)

4c

Prove or disprove: ker(T+S)⊆ker(T)⟹ker(S)⊆ker(T)Disproof:Let T=IVLet S=0VT+S=T⟹ker(T+S)=ker(T)⊆ker(T)ker(T)={0}ker(S)=Vdim(V)≥2⟹V⊈{0}

5

Let V be a finitely generated vector spaceLet T:V→V be a linear transformationU is called T-invariant if T[U]⊆UIn other words ∀u∈U:T(u)∈U

5a

Let T:V→V be a linear transformationLet U,W⊆V be T-invariant

5a.i

Prove: U+W is T-invariantProof:Let v∈U+W⟹∃u∈U,w∈W:v=u+w⟹T(v)=T(u+w)=T(u)⏟∈U+T(w)⏟∈W∈U+W⟹U+W is T-invariant

5a.ii

Prove: T[U]+T[W]=V⟹U+W=VProof:U⊆V,W⊆V⟹U+W⊆VLet v∈VT[U]+T[W]=V⟹∃u∈U,w∈W:T(u)+T(w)=vT(u)∈U,T(w)∈W⟹T(u)+T(w)∈U+W⟹v∈U+W⟹V⊆U+W⟹U+W=V

5b

Let T,S:V→V be linear transformationsLet TS=ST

5b.i

Prove: Im(S) is T-invariantProof:Let u∈Im(S)⟹∃v∈V:S(v)=uS(v)=u⟹T(S(v))=T(u)⟹S(T(v))=T(u)⟹T(u)∈Im(S)⟹Im(S) is T-invariant

5b.ii

Prove: ker(S) is T-invariantProof:Let v∈ker(S)⟹S(v)=0⟹T(S(v))=T(0)=0⟹S(T(v))=T(S(v))=0⟹T(v)∈ker(S)⟹ker(S) is T-invariant

5c

Let T:V→V be an isomorphismLet U⊆V be T-invariantProve: U is T−1-invariantProof:Let B={u1,u2,…,uk} be a basis of U⟹S={T(u1),T(u2),…,T(uk)}⊆ULet S be a linear dependence⟹∃i∈[1,k]:T(ui)∈sp(S∖{T(ui)})Let T(u1)∈sp(S∖{T(u1)})(WLOG)⟹∃{αj}j∈[2,k]:αjT(u1)=∑j=2kT(uj)⟹u1=T−1T(u1)=T−1(∑j=2kαjT(uj))=T−1T(∑j=2kαjuj)=∑j=2kαjuj⟹u1∈sp(B∖{u1})−Contradiction!⟹S is a linear independence⟹sp(S)=U⟹T[U]=ULet u∈UT[U]=U⟹∃v∈U:T(v)=u⟹T−1(u)=T−1(T(v))=v∈U⟹U is T−1-invariant