Linear Algebra, Theory And Applications, 2012a

9.5. EXERCISES 243
9. ↑In the situation of the above problem, let γ = {e 1 , ··· , e n } be the standard basis for
F n where e k is the vector which has 1 in the k th entry and zeros elsewhere. Show that
[T ] γ
=
(
u1 ··· u n
)
[T ]β
(
u1 ··· u n
) −1
(9.7)
10. ↑Generalize the above problem to the situation where T is given by specifying its
action on the vectors of a basis β = {u 1 , ··· , u n } as follows.
T u k =
n∑
a jk u j .
j=1
Letting A =(a ij ) , verify that for γ = {e 1 , ··· , e n } , (9.7) still holds and that [T ] β
= A.
11. Let P 3 denote the set of real polynomials of degree no more than 3, defined on an
interval [a, b]. Show that P 3 is a subspace of the vector space of all functions defined
on this interval. Show that a basis for P 3 is { 1,x,x 2 ,x 3} . Now let D denote the
differentiation operator which sends a function to its derivative. Show D is a linear
transformation which sends P 3 to P 3 . Find the matrix of this linear transformation
with respect to the given basis.
12. Generalize the above problem to P n , the space of polynomials of degree no more than
n with basis {1,x,··· ,x n } .
13. In the situation of the above problem, let the linear transformation be T = D 2 +1,
defined as Tf = f ′′ + f. Find the matrix of this linear transformation with respect to
the given basis {1,x,··· ,x n }. Write it down for n =4.
14. In calculus, the following situation is encountered. There exists a vector valued function
f :U → R m where U is an open subset of R n . Such a function is said to have
a derivative or to be differentiable at x ∈ U if there exists a linear transformation
T : R n → R m such that
|f (x + v) − f (x) − T v|
lim
=0.
v→0 |v|
First show that this linear transformation, if it exists, must be unique. Next show
that for β = {e 1 , ··· , e n } ,, the standard basis, the k th column of [T ] β
is
∂f
(x) .
∂x k
Actually, the result of this problem is a well kept secret. People typically don’t see
this in calculus. It is seen for the first time in advanced calculus if then.
15. Recall that A is similar to B if there exists a matrix P such that A = P −1 BP. Show
that if A and B are similar, then they have the same determinant. Give an example
of two matrices which are not similar but have the same determinant.
16. Suppose A ∈L(V,W) where dim (V ) > dim (W ) . Show ker (A) ≠ {0}. That is, show
there exist nonzero vectors v ∈ V such that Av = 0.
17. A vector v is in the convex hull of a nonempty set S if there are finitely many vectors
of S, {v 1 , ··· , v m } and nonnegative scalars {t 1 , ··· ,t m } such that
v =
m∑
t k v k ,
k=1
m∑
t k =1.
k=1