12-240/Classnotes for Thursday October 4

Recap

Base - what were doing today

Linear combination (lc) - We say v is a linear combination of a set S = {u1 ... un} if v = a1u1 ... anun for scalars from a field F.

Span - span(S) is the set of all linear combination of set S

Generate - We say S generates a vector space V is span(S) = V

Introduction to Basis

Linear dependance

Definition A set S subsetof V is called linearly dependant if you can express the zero vector as a linear combination of distinct vectors from S, excluding the non-trivial linear combination where all of the scalars are 0.

Otherwise, we call S linearly independant.

Examples

1. In R^3, take S = {u1 = (1,4,7), u2 = (2,5,8), u3 = (3,6,9)}

u1 - 2u2 + u3 = 0

S is linearly dependant.

2. In R^n, take {e_i} = {0, 0, ... 1, 0, 0, 0} where 1 is in the ith position, and (ei) is a vector with n entries.

Claim: This set is linearly independent.

Proof: Suppose (∑ ai*ei) = 0 ({ei} is linearly dependant.)

(∑ ai*ei) = 0 ⇔ a1(1, 0, ..., 0) + ... + an(0, ..., 0, 1) = 0 ⇔ (a1, 0, ... , 0) + ... + (0, ... , an) = 0

⇒ a1 = a2 = ... = an = 0!

Comments

1. {∅} is linearly independent. (Think about logical statements like "all elements of the empty set are purple") 2. {u} is linearly independent when u≠0.

Proof: