Results 1 to 5 of 5

Math Help - Vector Space

  1. #1
    Newbie
    Joined
    Feb 2011
    Posts
    2

    Vector Space

    What properties of vector space fail to hold the following:
    The set of all ordered triples of real numbers with the operations:
    U = (a,b,c), V = (x,y,z), r is a real number.
    Vector Addition: U + V = (a+x, b+y, c+z)
    Scalar Multiplication: r * U = (a, 1, c)

    I am fine with addition properties. They hold true. I am confused with scalar multiplication.

    Appreciate your help.
    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by Kandiah View Post
    What properties of vector space fail to hold the following:
    The set of all ordered triples of real numbers with the operations:
    U = (a,b,c), V = (x,y,z), r is a real number.
    Vector Addition: U + V = (a+x, b+y, c+z)
    Scalar Multiplication: r * U = (a, 1, c)

    I am fine with addition properties. They hold true. I am confused with scalar multiplication.

    Appreciate your help.
    Thanks.
    The scalar multiplication is not well defined Notice that

    2\vec{u}=[a,1,c] but

    2\vec{u}=\vec{u}+\vec{u}=[a+a,b+b,c+c]=[2a,2b,2c]
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Feb 2011
    Posts
    2
    Thanks for the response. Perhaps, since the scalar multiplication is not well defined I interpret differently and I may need further understanding. The scalar multiplication property is as follows: "If U (Vector) is in V (Vector Space) and c is a real number, then c*U is in V." Therefore, if a, b, & c in U are any real numbers and r is any real number, my interpretation is that r*U is not always equal to (a, 1, c). How do I interpret that r*U = (a, 1, C)?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by Kandiah View Post
    Thanks for the response. Perhaps, since the scalar multiplication is not well defined I interpret differently and I may need further understanding. The scalar multiplication property is as follows: "If U (Vector) is in V (Vector Space) and c is a real number, then c*U is in V." Therefore, if a, b, & c in U are any real numbers and r is any real number, my interpretation is that r*U is not always equal to (a, 1, c). How do I interpret that r*U = (a, 1, C)?
    There is not good way to interpret it. In the original post you ask what vector space axioms fail. Well scalar multiplication fails to distibute over additon.

    Vector space - Wikipedia, the free encyclopedia

    Here is a list of axioms. It is the 6th one in the list.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,368
    Thanks
    1313
    Another property that fails to hold here: In any vector space 0\vec{v}= \vec{0}. That is, the scalar, 0, times any vector is the 0 vector. By this definition of scalar multiplication, 0(a, b, c)= (a, 1, c)\ne (0, 0, 0) and (0, 0, 0) clearly is the additive identity.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Dual Space of a Vector Space Question
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 16th 2011, 03:02 AM
  2. Replies: 2
    Last Post: April 1st 2011, 02:40 AM
  3. Banach space with infinite vector space basis?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 24th 2011, 06:23 PM
  4. Replies: 15
    Last Post: July 23rd 2010, 11:46 AM
  5. Isomorphism beetwenn vector space and sub space
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 30th 2008, 10:05 AM

Search Tags


/mathhelpforum @mathhelpforum