Span of S over R^2

  • Thread starter AkilMAI
  • Start date
  • #1
AkilMAI
77
0

Homework Statement


Let S be the set of all vectors x = (x1; x2) in R^2 such that x1 = 1: What is the span of S ?


Homework Equations


.....


The Attempt at a Solution


x,y from S where x=(1,x2) ,y=(1,y2).Let w be the span of S => (w1,w2)=c1x+c2y......the system looks something like this w1=c1+c2 and w2=c1x1+c2y2...how can I find the 2 constants?
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,263
619
Can you show me a vector that's NOT in the span?
 
  • #3
AkilMAI
77
0
thanks again for the reply.....ok (w1,w2)=(w1-w2)x+w2y=>w1=w1-w2+w2 which is true but w2=w1x2-w2x2+w2y2
 
  • #4
Dick
Science Advisor
Homework Helper
26,263
619
thanks again for the reply.....ok (w1,w2)=(w1-w2)x+w2y=>w1=w1-w2+w2 which is true but w2=w1x2-w2x2+w2y2

I don't think you are thinking about this concretely enough. Is (2,5) in the span? Is (-1,1)? Is (2.67,10.32)? Is (0,1)? Is (1,0)?
 
  • #5
AkilMAI
77
0
wait...is span S=R^2?if so how can i prove it....the thing that I find confusing in the problem is the second coordonate of the each vector(ex. x2,y2......).
well (2,5)=c1x+c2y=>2=c1+c2,5=c1x2+c2y2...so if c1=c2=1 then x2=3 and y2=-3 .....hmm I'm not doing it right
 
  • #6
Dick
Science Advisor
Homework Helper
26,263
619
wait...is span S=R^2?if so how can i prove it....the thing that I find confusing in the problem is the second coordonate of the each vector(ex. x2,y2......).
well (2,5)=c1x+c2y=>2=c1+c2,5=c1x2+c2y2...so if c1=c2=1 then x2=3 and y2=-3 .....hmm I'm not doing it right

Yes, the span is R^2. (2,5)=(1,1)+(1,4). (1,1) and (1,4) are in S so (2,5) is in the span. Or (2,5)=2*(1,5/2) and (1,5/2) is in S. Do the other ones. It's good practice.
 
  • #7
AkilMAI
77
0
Ok,I understand ,thank you.....one last question...is there any way to prove it generally without the use of concrete examples?
 
  • #8
Dick
Science Advisor
Homework Helper
26,263
619
Ok,I understand ,thank you.....one last question...is there any way to prove it generally without the use of concrete examples?

If you prove (1,0) and (0,1) are in the span, that would prove it, wouldn't it?
 
  • #9
AkilMAI
77
0
Apparently I don't think I'm paying much attention to the problem.....I did prove (1,0) and are in the span (0,1)...but how will this prove for all x,y in R^2?
 
  • #10
AkilMAI
77
0
I mean,why just between the coordinates 0 and 1?
 
  • #11
Dick
Science Advisor
Homework Helper
26,263
619
I mean,why just between the coordinates 0 and 1?

The span of (1,0) and (0,1) is R^2, isn't it? I picked because that's a standard basis. (a,b)=a*(1,0)+b*(0,1). So if they are in the span then all of R^2 is in the span, right?
 
  • #12
AkilMAI
77
0
yes...
 

Suggested for: Span of S over R^2

Replies
1
Views
557
  • Last Post
Replies
10
Views
955
Replies
3
Views
1K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
2
Views
1K
Replies
6
Views
994
Replies
1
Views
2K
  • Last Post
Replies
8
Views
7K
  • Last Post
Replies
2
Views
5K
Top