Page 301 - 35Linear Algebra

16.4 Review Problems 301

Find the dimension of the kernel and image of this operator. What
happens if the target space is changed to P n−1 (x) or P n+1 (x)?

Now consider P 2 (x, y), the space of polynomials of degree two or less
in x and y. (Recall how degree is counted; xy is degree two, y is degree
2
one and x y is degree three, for example.) Let

∂ ∂
L := + : P 2 (x, y) → P 2 (x, y).
∂x ∂y

(For example, L(xy) = ∂ (xy) + ∂ (xy) = y + x.) Find a basis for the
∂x ∂y
kernel of L. Verify the dimension formula in this case.

7. Lets demonstrate some ways the dimension formula can break down if
a vector space is infinite dimensional.

(a) Let R[x] be the vector space of all polynomials in the variable x
with real coefficients. Let D = d be the usual derivative operator.
dx
Show that the range of D is R[x]. What is ker D?
n
Hint: Use the basis {x | n ∈ N}.
(b) Let L: R[x] → R[x] be the linear map

L(p(x)) = xp(x) .

What is the kernel and range of M?
(c) Let V be an infinite dimensional vector space and L: V → V be a
linear operator. Suppose that dim ker L < ∞, show that dim L(V )
is infinite. Also show that when dim L(V ) < ∞ that dim ker L is
infinite.

8. This question will answer the question, “If I choose a bit vector at
random, what is the probability that it lies in the span of some other
vectors?”

3
i. Given a collection S of k bit vectors in B , consider the bit ma-
trix M whose columns are the vectors in S. Show that S is linearly
independent if and only if the kernel of M is trivial, namely the
3
set kerM = {v ∈ B | Mv = 0} contains only the zero vector.

301