CHAPTER II DIMENSION In the present chapter we investigate ...

(f) If S 2 = O, then rank(S) ≤ nullity of S.
7. We are given matrix A and its reduced row echelon form B as follows:
⎡
2 4i 7i 3 2
⎤
0
⎡
1 2i 3i 1 1
⎤
0
⎢ 1
A = ⎣
0
2i
i
3i
5i
1
2i
2
−2
0 ⎥ ⎢ 0
⎦ =⇒ B = ⎣
0
0
1
0
5
1
2
−i
2i
2i
0 ⎥
⎦ .
0
1 2i 4i 2 0 0
0 0 0 0 0 0
(a) Write down a basis for the range of MA.
(b) Write down a basis for ker(MA).
(c) Write u = (1, 3i, 6i, −1 + 2i, 3, 0) as a linear combination of the row vectors of B.
(d) What is the rank and the nullity of MA?
8. For each linear map, find a basis of its kernel and a basis of its range.
(a) T : C3 → C2
−1 i 1 + i
given by T = MA, where A =
.
i 1 1 − i
(b) S : P3 → P2 given by T (p(x)) = xp ′′ (x) − 2p ′ (x) + p(1).
(c) Φ : M2,2 → M2,2 given by Φ(X) = AX − XA, where A =
1 3
.
0 5
9. Find a polynomial p(x) of degree 4 such that p(x + 1) − p(x) = x(x + 1)(x + 2). Then
use your result to give a formula for the sum n k= 0k(k + 1)(k + 2).
Exercises
1. Give a careful proof of the following statement: a set of vectors in V is a basis of V
if and only if it is linearly independent and it spans V .
2. Prove that if vectors v1, v2, . . . , v50 are linearly independent and are linear combinations
of vectors w1, w2, . . . , w50, then w1, w2, . . . , w50 are linearly independent.
3. Let T be a linear operator defined on a 2-dimensional real vector space such that
T 2 = −I. Show that there is a basis B of V such that [T ]B =
0 −1
1 0
4. Let T be a linear transformation from one vector space V into another W . Define a
new linear tranformation T2 : V × V → W by putting T2(x, y) = T x + T y. Suppose
that the rank of T is r and the dimension of V is n. What is the nullity of T2? Hint:
dim(V × V ) = 2n; T and T2 have the same range.
5. Let T be a linear transformation from a finite dimensional vector space V to another
W . Define a linear transformation T (2) : V → V × V by putting T (2) x = (T x, T x).
Show that T and T (2) have the same rank.
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.