Math 375 - 10/2

Question 1 • Is S={(█(1@2@−1@0)),(█(−1@−2@−1@0)),(█(π@√2@−1@1/2)),(█(−3@2@2@1)),(█(1@2@0@3))} independent? • Claim ○ If S is linearly dependent • Proof ○ If S is linearly independent, then § dim⁡(span(S))=|S|=5 ○ But because span(S) is a subspace of R4 § dim⁡(span(S))≤dim⁡〖R4 〗=4 ○ So S is linearly dependent Question 2 • Prove ○ 1,sin⁡x,sin⁡2x is linearly independent • Claim ○ ∀ a,b,c∈R ○ if a+b⋅sin⁡x+sin⁡2x=0, ∀x∈[0,1] ○ then a=b=c=0 • Proof ○ Set x=0 ⇒a=0 ○ Set x=π/6 ⇒ 1/2 b+√3/2 c=0 ○ Set x=π/4 ⇒b=c=0 ○ Therefore a=b=c=0