6.E: Exercises for Chapter 6
( \newcommand{\kernel}{\mathrm{null}\,}\)
Calculational Exercises
1. Define the map T:R2→R2 by T(x,y)=(x+y,x).
- Show that T is linear.
- Show that T is surjective.
- Find dim(null(T)).
- Find the matrix for T with respect to the canonical basis of R2.
- Find the matrix for T with respect to the canonical basis for the domain R2 and the basis ((1,1),(1,−1)) for the target space R2.
- Show that the map F:R2→R2 given by F(x,y)=(x+y,x+1) is not linear.
2. Let T∈L(R2) be defined by
T(xy)=(y−x), for all (xy)∈R2.
- Show that T is surjective.
- Find dim(null(T)).
- Find the matrix for T with respect to the canonical basis of R2.
- Show that the map F:R2→R2 given by F(x,y)=(x+y,x+1) is not linear.
3. Consider the complex vector spaces C2 and C3 with their canonical bases, and define S∈L(C3,C2) be the linear map defined by S(v)=Av,∀v∈C3, where A is the matrix
A=M(S)=(i112i−1−1).
Find a basis for null(S).
4. Give an example of a function f:R2→R having
the property that
∀a∈R,∀v∈R2,f(av)=af(v)
but such that f is not a linear map.
5. Show that the linear map T:F4→F2 is surjective if
null(T)={(x1,x2,x3,x4)∈F4 | x1=5x2,x3=7x4}.
6. Show that no linear map T:F5→F2 can
have as its null space the set
{(x1,x2,x3,x4,x5)∈F5 | x1=3x2,x3=x4=x5}.
7. Describe the set of solutions x=(x1,x2,x3)∈R3 of the system of equations
x1−x2+x3=0x1+2x2+x3=02x1+x2+2x3=0}.
Proof-Writing Exercises
1. Let V and W be vector spaces over F with V finite-dimensional, and let U be any
subspace of V . Given a linear map S∈L(U,W), prove that there exists a linear map
T∈L(V,W) such that, for every u∈U,S(u)=T(u).
2. Let V and W be vector spaces over F, and suppose that T∈L(V,W) is injective.
Given a linearly independent list (v1,…,vn) of vectors in V, prove that the
list (T(v1),…,T(vn)) is linearly independent in W.
3. Let U,V, and W be vector spaces over F, and suppose that the linear maps S∈L(U,V)
and T∈L(V,W) are both injective. Prove that the composition map T∘S is injective.
4. Let V and W be vector spaces over F, and suppose that T∈L(V,W) is surjective.
Given a spanning list (v1,…,vn) for V , prove that
span(T(v1),…,T(vn))=W.
5. Let V and W be vector spaces over F with V finite-dimensional. Given T∈L(V,W),
prove that there is a subspace U of V such that
U∩null(T)={0} and range(T)={T(u)|u∈U}.
6. Let V be a vector space over F, and suppose that there is a linear map T∈L(V,V)
such that both null(T) and range(T) are finite-dimensional subspaces of V . Prove that
V must also be finite-dimensional.
7. Let U,V, and W be finite-dimensional vector spaces over F with S∈L(U,V) and
T∈L(V,W). Prove that
dim(null(T∘S))≤dim(null(T))+dim(null(S)).
8. Let V be a finite-dimensional vector space over F with S,T∈L(V,V). Prove that
T∘S is invertible if and only if both S and T are invertible.
9. Let V be a finite-dimensional vector space over F with S,T∈L(V,V), and denote by
I the identity map on V . Prove that T∘S=I if and only if S∘T=I.
Contributors
- Isaiah Lankham, Mathematics Department at UC Davis
- Bruno Nachtergaele, Mathematics Department at UC Davis
- Anne Schilling, Mathematics Department at UC Davis
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