Important Definitions and Theorems
Test Wed. 3/28
Text Sections for the test:
2.2, 3.1 to 3.5
Most definitions and theorems from Chapter 3 are given below. Be sure you also study
2.2, especially # 13, 14 from p. 109.
Def. A subspace of a vector space V is a subset W of V which is a vector space under the
inherited operations from V.
Theorem 3.3: W is a subspace of V if and only if the W is closed under vector addition
and under scalar multiplication, i.e.,
a) For all u,v in W, u + v is in W.
b) For all u in W and for r any real number, r ∙ u is in W.
Def. A linear combination of vectors v1, ... ,vn is a sum of the form
a1v1 + ... + anvn
For the definitions below, let V be a vector space and let S = { v1, ... ,vn } be a set of
vectors in V.
Def. S spans V if every vector in V can be written as a linear combination of vectors in S,
i.e., if the system
a1v1 + ... + anvn = v
is consistent for every possible vector v in V.
(We also say in this case, “S is a spanning set for V.”)
Def. The span of S, also written span (S), is the set of all linear combinations of vectors
in S.
Def. S is linearly independent if whenever
a1v1 + ... + anvn = 0
then a1 = ... = an = 0
S is dependent if there is a solution to the above system in which some aj ≠ 0.
Def. S is a basis for V if 1) S spans V; 2) S is linearly ind.
Theorem Every basis for V has the same number of vectors, assuming V has some finite
basis.
Def. The dimension of V is the number of vectors in a basis for V.
Theorem 3.6: Assume all vectors in S are non-zero and S is non-empty. Then S is
dependent if and only if some vector in S can be written as a linear combination of the
other vectors in S.
Definitions,Math 310, page1
Theorem: If S consists of exactly two vectors, then S is independent if and only if the
vectors are not scalar multiples of each other. (Note: this theorem is given as exercise 18
on p. 163.)
Theorem 3.8 (restriction): If S spans V, then there is a subset T of S such that T is a
basis for V.
Theorem 3.5: Suppose S and T are sets in V such that S is a subset of T.
a) If S is lin. dep., then so is T.
b) If T is lin. ind., then so is S.
c) If S spans V, then so does T.
Corollary 3.4: If the dimension of V is n, then any set of m > n elements in V is lin. dep.
Corollary 3.5:
span V.
If the dimension of V is n, then any set of m < n elements in V cannot
Theorem 3.11: Suppose dim V = n.
a) Any set of n vectors in V which is lin. ind. is a basis for V.
b) Any set of n vectors in V which spans V is a basis for V.
Theorem 3.11 can also be phrased as:
“Suppose dim V = n. Then a set of n vectors is ind. if and only if it spans V.”
Definitions,Math 310, page2