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

(c) u + iv, iu + v are linearly independent.
4. True or False:
(a) A set S consisting of a single vector, say S = {v}, is always linearly independent.
(b) If two vectors are linearly dependent, then one of them is a scalar multiple of the
other.
(c) If two vectors are linearly dependent, then each of them is a scalar multiple of
the other.
(d) If v1, v2, v3 are linearly dependent, then so are v1, v2.
(e) A set of vectors is linearly independent if and only if none of them is in the span
of the rest of them.
(f) A set of three vectors is linearly independent if each pair of them is a linearly
independent set.
5. Prove the following two statements:
(a) 2×2 matrices A1, A2, A3 (considered as vectors in M2,2) are linearly independent
if BA1, BA2, BA3 are linearly independent, where B is some 2 × 2 matrix.
(b) Polynomials p1(x), p2(x), p3(x) (considered as vectors in P) are linearly independent,
if their derivatives p ′ 1 (x), p′ 2 (x), p′ 3 (x) are linearly independent.
6. In each of the following cases, find the leading vectors of the given list L of vectors
in a linear space V and express the other vectors in the list as linear combinations of
the leading vectors.
(a) V = P2, L = x + 1, 2x + 2, x − 1, 2x, x 2 − 2x, x 2 + 5x + 1
(b) V = C 2 , L = ( (1 + i, 1 − i), (i, 1), (1, i), (1, 2) )
(c) V = C 3 , L = ( C1, C2, C3, C4, C5, C6 ), assuming that the matrix
C = [C1 C2 C3 C4 C5 C6]
has the following reduced row echelon form:
⎡
1
R = ⎣ 0
2
0
0
1
0
0
5
2
⎤
3
4 ⎦
0 0 0 1 6 7
7. verify that subspaces V1, V2, V3 in Example 1.3.1 are invariant for both operators
D and Ta.
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