Recitation 1 Lecturer : Maurice Cheung Topic : Linear Algebra Review 1 Linear Algebra Review

1.1 Independence, Spanning, and Dimension Definition 1 A (usually infinite) set of vectors S is a vector space if ∀x, y ∈ S, λ ∈ R, (a) x + y ∈ S (b) λx ∈ S, and (c) 0 ∈ S. Definition 2 A set of vectors x1, . . . , xk is said to be linearly dependent if there exists a vector λ 6= 0 such that ki=1 λix = 0. Otherwise, the set is linearly independent. Claim 1 If S is linearly dependent and S ⊂ T , then T is also linearly dependent. If some set S is linearly independent and S ⊃ T , then T is also linearly independent. Proof: If T = S, we’re done. Otherwise, WLOG T = x1, . . . , xk and S = x1, . . . , xl where l < k. Since S is linearly dependent, ∃λ 6= 0, λ ∈ Rl such that li=1 λix = 0. However, letting λl+1 = · · · = λk = 0 gives ∑k i=1 λix i = 0. Thus, T is linearly dependent. The second statement is just the contrapositive. 2