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Questions about linear independent and spanning set and basi

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  1. Jan 5, 2016 #1
    • Member warned about posting with no effort shown
    1. The problem statement, all variables and given/known data

    Definition of spanning set:
    Let be vectors in the vector space . The set of all linear combination of the vectors is a subspace ( say ) of . The subspace is called the space spanned by the vectors The set is called a spanning set of .

    Definition of linear independence:
    Suppose that is a vector space. The set of vectors from is linearly dependent if there is a relation oflinear dependence on that is not trivial. In the case where the only relation of linear dependence on is the trivial one, then is a linearly independent set of vectors.

    Definition of basis of vector space:
    1. It spans the space.
    2. Its vectors are independent.
    3. The number of vectors in the basis is equal to the dimension of the space.



    True or false? Given reason

    1. A set of 5 Vectors in R5 must be a basis for R5
    2. A set of 6 Vectors in R5 cannot be a basis for R5
    3. A set of 7 vectors in R5 must be a spanning set for R5
    4. A set of 6 polynomials in R5 must be a basis for R5
    5. A set of 6 polynomials in R5 may be a basis for R5


    Anyone help me to explain these question.. I want to know the reason cz it's confused me a lot and i always get mess about these kind of question
    appericated it!
     
    Last edited: Jan 5, 2016
  2. jcsd
  3. Jan 5, 2016 #2

    Samy_A

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    I assume that what you wrote under "Relevant equations" is actually the "The problem statement", and that R5 stand for ##\mathbb R^5##.

    For starters, maybe you could define the used concepts: basis and spanning set.
     
  4. Jan 5, 2016 #3
    Thanks. I just edited the form of the question and now it should be ok..btw, Can u help me to this question?
     
  5. Jan 5, 2016 #4

    Samy_A

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    Thanks.

    The forum rules expect you to show some effort in solving the exercise, before other forum members jump in to help.
    That's why I suggested you at least start with defining what a basis is, and what a spanning set is.
     
  6. Jan 5, 2016 #5
    Thanks for reminding. This is my first time to ask question here so sorry about that.
     
  7. Jan 5, 2016 #6

    Samy_A

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    (I added the bold blue)
    Using that third point, can you answer some of the questions?
     
  8. Jan 5, 2016 #7
    I can answer question 2 the reason is any set of 6 vectors in R5 is linearly dependent, even though some sets of 6 vectors in R5 span R5. To be a basis it must be a linearly independent spanning set, so if it's linearly dependent, it cannot be a basis.
    But actually I know the answer for all of these questions but i don't the reason apart from question2
     
  9. Jan 5, 2016 #8

    Samy_A

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    Ok. What about questions 4 and 5?
     
  10. Jan 5, 2016 #9
    No i don't know how to explain question 1 3 4 5.
     
  11. Jan 5, 2016 #10

    Samy_A

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    Let's take 4 as an example:
    "4. A set of 6 polynomials in R5 must be a basis for R5"

    Can a set of 6 polynomials be a basis for ##\mathbb R^5##? The answer can be found using the definitions you posted.
     
  12. Jan 5, 2016 #11
    Thanks for helping. Here is too late now so I have to sleep. I am gonna do these question tomorrow.
    Appericated for helping!!
     
  13. Jan 5, 2016 #12

    Mark44

    Staff: Mentor

    Noted...
     
  14. Jan 7, 2016 #13
    Hi guys. I hv done these question already. Thanks for helping!!
     
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