PROBLEM SET I
DUE FRIDAY,  FEBRUARY
Exercise . Suppose V and W are vector spaces. Show that the set Lin(V, W) of
linear maps V . W is a vector space, where addition and scalar multiplication
are de ned by the formulas
(ϕ + ψ)(v) = ϕ(v) + ψ(v),
(αϕ)(v) = α(ϕ(v)),
where ϕ, ψ ∈ Lin(V, W) and α ∈ R. If V is of dimension m and W is of
dimension n, then what is the dimension of Lin(V, W)?
De nition. e dual space of a vector space V is the space V ∨ := Lin(V, R).
Exercise . Suppose V is a vector space of dimension m, and suppose W ⊂ V
is a vector subspace of dimension n. Consider the following subset of V ∨ :
Ann(W) := {ϕ ∈ V ∨ | W ⊂ ker ϕ}.
Show that Ann(W) is a vector subspace of dimension m − n.
Exercise . Suppose U ⊂ R an open set, and consider the set C0 (U) of continuous functions on U and the set C1 (U) of continuously differentiable functions
on U. Equipped with the usual notion of addition and scalar multiplication of
functions, these are vector spaces. Show that the function
d : C1 (U) .
C0 (U)
given by the formula d(g) := g ′ is a linear map. Compute its kernel and its
image.
Exercise⋆ . Suppose α ∈ R. Show that the set S(α) of all functions g ∈
C∞ (R) that satisfy the differential equation
g ′′ + αg = 0
is a vector space (with the usual notion of addition and scalar multiplication of
functions). Compute the dimension of S(α). Does it depend on α?
Suppose now that m is a positive integer and that
σ : {1, 2, . . . , m} .

{1, 2, . . . , m}

DUE FRIDAY,  FEBRUARY
is any bijection. De ne a linear map Tσ : Rm .
Rm by de ning
Tσ (ei ) := eσ(i) ,
and extending by linearity.
Exercise . Show that Tσ is an isomorphism.
Exercise . Compute the matrix corresponding to Tσ , and write an explicit
formula for the vector Tσ (v) for any vector v = (x1 , x2 , . . . , xm ) ∈ Rm .
Exercise⋆ . Show that the span of the set of linear maps Tσ is the set of linear
maps Rm . Rm whose corresponding matrices M have the following property:
there exists a real number r such that the sum of the entries in each row is r,
and the sum of the entries in each column of M is also r.