Proving Equality of Spans of Vectors in Rn

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In summary, Two homework problems are given. The first problem asks to prove the equality of the spans of two sets of vectors using the definition of span. The second problem asks to show that one span is contained in another span using only the definition of span. In the first problem, starting with a vector w in the span of {u, v}, it is shown that w can also be written as a linear combination of u + v and u - v. In the second problem, the goal is to show that any vector in the span of S1 is also in the span of S2. This can be done by showing that any linear combination of vectors in S1 can also be written as a linear combination of vectors in S2
  • #1
mat175
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Homework Statement



1. let u and v be any vectors in Rn. Prove that the spans of {u,v,} and {u+v, u-v} are equal.

2. Let S1 and S2 be finite subsets of Rn such that S1 is contained in S2. Use only the definition of span s1 is contained in span s2.

Homework Equations


The Attempt at a Solution



1. w in the span (u+v, u-v) show that w is in the span (u,v)
w is a linear combination of (u+v, u-v)
w= c1 (u+v) + c2(u-v)------
w=(c1+c2)u+(c1-c2)v

How do you prove it going the other way though?


2. I am not sure how to start this one, other than having the definition of span there.
 
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  • #2
For 1, start with a vector w that is in the span of {u, v}, which means that w = c1u + c2v. See if you can get creative on the two constants to rewrite them as the sum of two numbers and the difference of two numbers, respectively. If haven't done this, but that's the direction I would take.

For 2, you are missing some words in the problem statement. I'm guessing that this is the actual problem statement: "Use only the definition of span to show that span s1 is contained in span s2." Please confirm that this is the correct interpretation.
 
  • #3
Yes that is correct.

For 1, I did have what you have then I don't know for sure where to go from there.

Thanks.
 
  • #4
mat175 said:
Yes that is correct.

For 1, I did have what you have then I don't know for sure where to go from there.

Thanks.
Show me what you've done for the 2nd half of this problem.
 
  • #5
All I have is let y be a linear combination of (u,v)
which is written y=c1u1+c2u2
 
  • #6
mat175 said:
All I have is let y be a linear combination of (u,v)
which is written y=c1u1+c2u2
Well, no. It would be y = c1u + c2v, right? You want y to be a linear combination of u and v, not u1 and u2.

Starting with y as I have written it, you want to end up with y (the same y) as a linear combination of u + v and u - v, right? What does that look like?

Now you know what you're starting with, and you sort of know what you want to end up with. Can you work backward from the end to the beginning and fill in the middle?
 

1. What is the definition of "span" in the context of vectors in Rn?

In linear algebra, the span of a set of vectors is the set of all linear combinations of those vectors. In other words, it is the set of all possible vectors that can be created by multiplying each vector by a constant and adding them together.

2. How can I prove that two spans of vectors in Rn are equal?

To prove equality of spans, you need to show that each span contains all the same vectors as the other. This can be done by showing that every vector in one span can be written as a linear combination of the vectors in the other span, and vice versa.

3. What is the significance of proving equality of spans in Rn?

Proving equality of spans is important because it allows us to show that two sets of vectors have the same "spanning power", meaning they can both create the same set of vectors through linear combinations. This can be useful in various applications, such as solving systems of linear equations or determining the dimension of a vector space.

4. Can you give an example of proving equality of spans in Rn?

Yes, for example, if we have two sets of vectors in R3: S = {(1,0,0), (0,1,0)} and T = {(1,1,0), (0,1,1)}, we can prove that the span of S is equal to the span of T by showing that each vector in S can be written as a linear combination of the vectors in T (and vice versa). In this case, we can write (1,0,0) = (1,1,0) - (0,1,0) and (0,1,0) = (0,1,1) - (0,0,1), thus proving equality of spans.

5. Are there any shortcuts or tricks for proving equality of spans in Rn?

Yes, there are several methods that can be used to make proving equality of spans easier, such as using linear independence, Gaussian elimination, or matrix operations. It is important to choose the method that best fits the specific problem at hand.

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