Description
1. Consider the set of all n × n real matrices. This set is has a vector
space structure, as we have seen in class. Prove that
S = {A ∈ R
n×n
|A
T = −A}
that is the set of all skew symmetric matrices, is a subspace. Here, we
have denoted by AT
the transpose of A, that is the matrix {a
T
ij} = AT
defined by a
T
ij = aji.
2. Consider T : P3 → P2 defined by differentiation, i.e., by T(p) = p
0 ∈ P2
for p ∈ P3. Find the range and the Null space for T.
3. Let A be a n × n matrix with real coefficients and let TA : R
n → R
n
denote the linear operator defined by
TAx = A · x,
for every x ∈ R
n
. Prove that R(TA) is equal to the span of the columns
of A.
4. Let T : R
n → R
n be a linear operator and for every k ∈ N set T
k
to
denote the composition of T with itself k times.
(i) Show that for every k ∈ N one has R(T
k+1) ⊂ R(T
k
).
(ii) Show that there exists a positive integer m such that for all k ≥ m
one has R(T
k
) = R(T
k+1).
5. Let A and B be two square, n × n matrices. Prove that if AB = 0 (as
matrix products), then
R(TA) + R(TB) ≤ n,
where we have denoted by TA and TB the linear operators associated
to the matrices A and B respectively.
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