Which of the following statements are true/false?Justify your answer. (i) R2 has infinitely many non-zero,...

Trace Cline

Trace Cline

Answered

2022-01-22

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.

Answer & Explanation

lorugb

lorugb

Expert

2022-01-23Added 13 answers

(i) We can construct such a set of subspaces:
1) rR, let: Vr={(x,rx)R2xR}.
[Geometrically, Vr is the line through origin of R2, os slope r.]
2) We will check that these subspaces justify assertion (i).
3) Clearly: VrR2.
4) Check that: Vr is a proper subspace of R2..
Let: u,vVr,α,βR. Verify that: αu+βvVr
u,vVr,u=(x1,rx1),v=(x2,rx2); for some x1,x2R
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,vVr,α,βRαu+βvVr.
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) r1=r+1
r=r+1, plainly impossible.
Thus: Vr is a non-zero, proper subspace of

Tapanuiwp

Tapanuiwp

Expert

2022-01-24Added 13 answers

(i) True.
(ii) False.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?