Important Definitions and Theorems

Important Definitions and Theorems

Test Wed. 4/30

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 =

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 4.7: 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.

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.

Theorem 4.5: Assume S is a set of n vectors in Rn. Then S is linearly independent if and only if det A ≠ 0.

Theorem 4.6: 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.

Theorem 4.9 (restriction): If S spans V, then there is a subset T of S such that T is a basis for V.

Corollary 4.4: If the dimension of V is n, then any set of m > n elements in V is lin. dep.

Corollary 4.5: If the dimension of V is n, then any set of m < n elements in V cannot span V.

Theorem 4.12: 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 4.12 can also be phrased as:

“Suppose dim V = n. Then a set of n vectors is ind. if and only if it spans V.”

Theorem:

Suppose A is an n x n matrix. Then the following are equivalent:

a)  A is nonsingular.

b)  A is row equivalent to In.

c)  det A ≠ 0,

d)  The columns of A are linearly ind.

e)  The rows of A are linearly ind.

f)  The columns of A span Rn.

g)  The columns of A form a basis for Rn.

Definitions,Math 310, page1