4.3 Flashcards
Theorem 4.3.1…
A set S with 2 or more vector is
a) Linearly dependent if and only if at least one of the vectors in S is expressible as a linear combination of the other vectors in S.
b) Linearly independent if and only if no vector in S is expressible as a linear combination of the other vectors in S.
- 3.2: a) a finite set that contains the zero vector is linearly dependent
b) a set with exactly one vector is linearly independent if and only if that vector is not the zero vector.
c) A set with exactly 2 vectors is linearly independent if and only if neither vector is a scalar multiple of the other.
proof of a: for any vector v1, v2, …, vr the set S = (v1, v2, …, vr, 0 ) is linearly dependent since the equation:
0v1 + 0 v2 + … + 0 vr + 1 (0) = 0
10 Show that if (v1, v2, v3) is a linearly independent set of vectors then so are (v1, v2), (v1, v3), (v2, v3), (v1), (v2) and (v3).
v1, v2, v3 are linearly independent
therefore, none of v1, v2 and v3 is the zero vector. so each v1, v2 abd v3 individually are linearly independent
suppose c1v1 + c2v2 = 0 then c1v1 + c2v2 + 0 v3 = 0 thus c1 = c2 = 0 since they are linearly independent. this proves that (v1, v2) is a linearly independent set. similarly for (v1, v3) and (v2, v3).
11 Show that if S = (v1, v2, …, vr) is a linearly independent set of vectors, then so is every nonempty subset of S.
suppose (v1, v2, …, vr) is a linearly independent set and suppose u1, u2,…, um is a subset that is to say each ui is a vi.
suppose k1u1 + k2u2 + … + kmum = 0
if c1v1 + c2v2 + … + crvr = 0 then c1 = c2 = c3 = … = cr = 0
k1u1 + k2u2 + 0um+1 + 0um+2 + … + 0um-r = 0
therefore, k1 = k2 = … = km = 0
12 show that if S = (v1, v2, v3) is a linearly dependent set of vectors in a vector space V, and v4 is any vector in V that is not in S, then (v1 , v2 ,v3 , v4) is also linearly dependent.
know to do it.
17 The space spanned by 2 vectors in r3 is a line though the origin, a plane through the origin or the origin itself.
know to do it.
