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Math Help - Question about vectors and dimensions ?

  1. #1
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    Question about vectors and dimensions ?

    Let V be a vector space and S = { v1, ... , vn } a set of vectors from V.

    1) What conditions do the vectors from S need to satisfy in order for them to form a basis for V ?

    2) What would be the dimension of V in this case ?

    3) Find the dimension and a basis for the linear subspace R^3 spanned by the vectors:
    (3,-2,4) (1,-1,0) (3,-1,8)

    4) Show that the vectors (1,1,0) (1,0,1) (0,1,1) form a basis for R^3

    5) Find the components of the vector (3,4,5) in the basis considered in 4

    this is no homework, this is a past exam papers without answers, i could really use this answers to revise, thank you in advance !
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by abrahamtim View Post
    Let V be a vector space and S = { v1, ... , vn } a set of vectors from V.

    1) What conditions do the vectors from S need to satisfy in order for them to form a basis for V ?
    They would need to be linearly independednt and span the space.

    2) What would be the dimension of V in this case ?
    Well what do your notes say?

    3) Find the dimension and a basis for the linear subspace R^3 spanned by the vectors:
    (3,-2,4) (1,-1,0) (3,-1,8)
    Well are they linearly independednt? If no what is the largest subset of them which is linearly independednt?

    4) Show that the vectors (1,1,0) (1,0,1) (0,1,1) form a basis for R^3
    You know what you have to do, show they are linearly independednt and that they span R^3

    5) Find the components of the vector (3,4,5) in the basis considered in 4

    this is no homework, this is a past exam papers without answers, i could really use this answers to revise, thank you in advance !
    CB
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  3. #3
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    Quote Originally Posted by abrahamtim View Post
    Let V be a vector space and S = { v1, ... , vn } a set of vectors from V.

    1) What conditions do the vectors from S need to satisfy in order for them to form a basis for V ?

    2) What would be the dimension of V in this case ?
    These two are basic definitions- they should be the first thing you learn. The dimension of a space is defined as the number of vectors in a basis.

    3) Find the dimension and a basis for the linear subspace R^3 spanned by the vectors:
    (3,-2,4) (1,-1,0) (3,-1,8)
    As Captain Black said, are they independent?

    4) Show that the vectors (1,1,0) (1,0,1) (0,1,1) form a basis for R^3
    Again, are they independent? Do they span R^3.

    5) Find the components of the vector (3,4,5) in the basis considered in 4
    In other words you want to find numbers, a, b, c, such that a(1, 1, 0)+ b(1, 0, 1)+ c(0, 1, 1)= (3, 4, 5). That is, you must have a+ b= 3, a+ c= 4, b+ c= 5. Solve those equations.

    this is no homework, this is a past exam papers without answers, i could really use this answers to revise, thank you in advance !
    To revise? If you are reviewing a course you have already taken, you certainly should know basic definitions!
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