Span of S over R^2: Find c1,c2

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In summary, the problem asks what the span of the set S, which consists of all vectors in R^2 with a first coordinate of 1, is. The solution involves finding two constants, c1 and c2, such that w1 = c1 + c2 and w2 = c1x2 + c2y2. By finding the values of these constants, it can be shown that any vector in R^2 can be written as a linear combination of vectors in S, thus proving that the span of S is R^2.
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AkilMAI
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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


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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?
 
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  • #2
Can you show me a vector that's NOT in the span?
 
  • #3
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
AkilMAI said:
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
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
AkilMAI said:
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
Ok,I understand ,thank you...one last question...is there any way to prove it generally without the use of concrete examples?
 
  • #8
AkilMAI said:
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
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
I mean,why just between the coordinates 0 and 1?
 
  • #11
AkilMAI said:
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
yes...
 

1. What is the span of S over R^2?

The span of S over R^2 is the set of all possible linear combinations of the vectors in S, where the coefficients are real numbers and the resulting vectors are in the two-dimensional space R^2.

2. How do you find the span of S over R^2?

To find the span of S over R^2, you need to determine the values of c1 and c2 that satisfy the equation c1v1 + c2v2 = w, where v1 and v2 are the vectors in S and w is any vector in R^2. This will give you the range of possible values for c1 and c2 that make up the span.

3. What is the significance of the span of S over R^2?

The span of S over R^2 is a fundamental concept in linear algebra and is used to understand and solve systems of linear equations. It helps to determine the solutions to these equations and also gives insights into the nature of vector spaces.

4. Can the span of S over R^2 be empty?

No, the span of S over R^2 cannot be empty. Since the vectors in S are in R^2, any linear combination of these vectors will also be in R^2. Therefore, the span will always contain at least the zero vector, making it a non-empty set.

5. How does the span of S over R^2 relate to linear independence?

The span of S over R^2 is closely related to the concept of linear independence. If the vectors in S are linearly independent, then the span will be the entire space R^2. However, if the vectors are linearly dependent, then the span will be a smaller subspace of R^2.

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