I have a midterm tomorrow, just a conceptual question needs to be answered immediatel

In summary, the discussion revolved around the concept of span and whether two given vectors, v1 and v2, can span a plane in R2. The conclusion was that if v1 is not a scalar multiple of v2, then they will span a plane defined by all combinations cv1+dv2, where c and d are real numbers. However, whether this plane is R2 or not depends on the vectors and what one considers to be R2. Additionally, the concept of "free variables" or "parameters" was discussed in relation to the number of vectors and the dimension of the span.
  • #1
flyingpig
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1

Homework Statement



I am just wondering if I have Span{v1,v2} where v1 is not a scalar multiple of v2, then it is known that they span a plane, in fact it is R2

So the question is, how? There can be at most one parameter right? One parameter means the solution is a line, we need two to make a plane?
 
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  • #2


if the span were just a line, then v2 would have to be a scalar multiple of v1, and that the line would basically be equivalent to all scalar multiples of just one of the vectors. i.e. if it this were the case, we would have span{v1}=span{v2}=span{v1, v2}. since v1 is not a scalar multiple of v2 in your case, it is impossible that they lie on the same line, thus they span a plane defined by all combinations cv1+dv2, where c and d are real numbers. Also, you have provided nothing that would suggest the vectors are two dimensional, so I wouldn't say they span R2. They span some plane, yes, but not necessarily R2
 
  • #3


I thought Span is the linear combination of all vectors in that set and hence it must be R^2?
 
  • #4


flyingpig said:
I thought Span is the linear combination of all vectors in that set and hence it must be R^2?

your set consists of two vectors, v1 and v2. if v1 and v2 are both elements of R^2, then they will span all of R^2. nowhere have you stated that v1 and v2 are both elements of R^2, they could be vectors in R^3.
 
  • #5


Theorem. said:
your set consists of two vectors, v1 and v2. if v1 and v2 are both elements of R^2, then they will span all of R^2.
Assuming v1 and v2 are in R^2, if v1 = c v2 for some c in R, then no, span{v1, v2} is not equal to R^2.
 
  • #6


Unit said:
Assuming v1 and v2 are in R^2, if v1 = c v2 for some c in R, then no, span{v1, v2} is not equal to R^2.

The OP specificially stated that v1 is not qual to a scalar multiple of v2
 
  • #7


Theorem. said:
your set consists of two vectors, v1 and v2. if v1 and v2 are both elements of R^2, then they will span all of R^2. nowhere have you stated that v1 and v2 are both elements of R^2, they could be vectors in R^3.

Does it matter how many entries I have in my vectors? I can have three entries in my vectors, but with two vectors, they will always span R2.

Is there even any relations to having free variables (parameters) with the concept of Span?
 
  • #8


My apologies, Theorem; I was hasty. You are right.
 
  • #9


Unit said:
My apologies, Theorem; I was hasty. You are right.

No worries : ) hopefully the OP understands what i meant
 
  • #10


Theorem. said:
No worries : ) hopefully the OP understands what i meant

I don't unfortunately =(
 
  • #11


Please I got to sleep soon lol
 
  • #12


flyingpig said:
Does it matter how many entries I have in my vectors? I can have three entries in my vectors, but with two vectors, they will always span R2.

Is there even any relations to having free variables (parameters) with the concept of Span?
Two independent vectors will always span a plane. Whether or not that plane is R2 or not depends upon the vectors and what you mean by R2. The two vectors <1, 0, 1> and <0, 1, 0> will have span a<1, 0, 1>+ b<0, 1, 0>= <a, b, a> which can be expressed as <x, y, z> with z= x, or the plane z= x.
If the two vectors are <1, 0, 0> and <0, 1, 0> then they span <1, 1, 0>, the xy- plane. Do you consider that to be R2? Some people think of R2 as a subset of R3, some do not.

I'm not sure what you mean by "free variables (parameters) with the concept of Span" but in your first post you asked:
So the question is, how? There can be at most one parameter right? One parameter means the solution is a line, we need two to make a plane?
The definition of "span" of a set of vectors is the set of all linear combinations of the vectors. In particular, the span of the two vectors {u, v} is all vectors of the form au+ bv for any scalars a and b. Both a and b are, I think, what you are calling "parameters". Where did you get the idea that there could be "at most one parameter"? The span of a set of n vectors will involve n parameters and, if the vectors are independent, the span will be an n-dimensional subspace.
 

1. What type of conceptual question might I encounter on my midterm?

Conceptual questions on a midterm can cover a wide range of topics, but they typically require you to apply your understanding of a concept or theory to a specific scenario or problem.

2. How should I prepare for a conceptual question on my midterm?

To prepare for a conceptual question, make sure you have a solid understanding of the main concepts and theories covered in your course. Review your notes, textbook, and any practice problems or quizzes to help you identify key ideas and how they can be applied.

3. Will the conceptual question on my midterm be multiple choice or open-ended?

This can vary depending on the instructor and the specific question, but many conceptual questions on midterms are open-ended or require some written explanation. It's important to read the question carefully and answer it in a way that fully demonstrates your understanding.

4. Can I use outside resources to answer a conceptual question on my midterm?

It's best to check with your instructor beforehand, but in most cases, you will be expected to answer conceptual questions on your own without consulting outside resources. This allows your instructor to assess your understanding of the material and your ability to apply it.

5. How much time should I spend on a conceptual question during my midterm?

This can vary depending on the length and complexity of the question, but it's important to manage your time effectively during a midterm. You may want to allocate a set amount of time to each question and move on if you are struggling, then come back to it later if time allows.

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