Which of the following statements are true/false?Justify your answer. (i) R2 has infinitely many non-zero, proper vector subspaces. (ii) Every system of homogeneous linear equations has a non zero solution.

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Which of the following statements are true/false?Justify your answer.   (i) R2 has infinitely many non-zero, proper vector subspaces.  (ii) Every system of homogeneous linear equations has a non zero solution.

lorugb · Accepted Answer

(i) We can construct such a set of subspaces: 1) ∀r∈R, let: Vr={(x,rx)∈R2∣x∈R}. [Geometrically, Vr is the line through origin of R2, os slope r.] 2) We will check that these subspaces justify assertion (i). 3) Clearly: Vr⊆R2. 4) Check that: Vr is a proper subspace of R2.. Let: u,v∈Vr,α,β∈R. Verify that: αu+βv∈Vr u,v∈Vr,⇒u=(x1,rx1),v=(x2,rx2); for some x1,x2∈R ∴αu+βv=α(x1,rx1)+β(x2,rx2) =α(x1,rx1)+β(x2,rx2) =(αx1,αrx1)+(βx2,βrx2) =(αx1+βx2,αrx2+βrx2) =(αx1+βx2,r(αx1+βx2) =(x3,rx3)∈Vr; with x3=αx1+βx2 So: u,v∈Vr,α,β∈R⇒αu+βv∈Vr. Thus: Vr is a subspace of R2 To see that Vr is non-zero, note that: (1,r)∈Vr and (1,r)≠(0,0). To see that Vr is proper, note that (1,r+1)∉Vr: (1,r+1)∈Vr⇒ (by construction of Vr) r⋅1=r+1 ⇒r=r+1, plainly impossible. Thus: Vr is a non-zero, proper subspace of

Tapanuiwp · Accepted Answer

(i) True.    (ii) False.

Which of the following statements are true/false?Justify your answer. (i)

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