1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Urgent: Linear Algebra Question(Please verify)

  1. Oct 8, 2006 #1
    (a)

    show that the vector (2,7,6) be we written as a linear combination of the vectors

    (1,3,2) and (0,1,2)

    (b) show that the vector (-1,0,4) can be written as a linear combination of the vectors (1,3,2) and (0,1,2)

    (c)

    show that Span((1,3,2),(0,1,2)) = Span( (2,7,6), (-1,0,4))

    My solution (a).

    I write vectors in equation form

    [tex]x_1 \[ \[ \begin{array}{c} 1 \\ 3 \\ 2 \end{array} \] + x_2 \[ \begin{array}{dd} 0 \\ 1 \\ 2 \end{array} \] =
    \[ \begin{array}{c} 2 \\ 7 \\ 6 \end{array} \] [/tex]

    which can be rewritten to

    [tex] \[ \begin{array}{ccc} x_{1} \\ 3x_{1} + x_{2} \\ 2x_{1} + 2x_{2} \end{array} \] = \[ \begin{array}{c} 2 \\ 7 \\ 6 \end{array} \][/tex]

    There must exist x_1 and x_2 which makes the above set of equations true.

    These are found be rewritten the system into its equivalent coefficient matrix.

    [tex] \[ \begin{array}{ccc} 1 & 0 & 2 \\ 3 & 1 & 7 \\ 2 & 2 & 6 \end{array} \] [/tex]

    using row reduction I get

    [tex] \[ \begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{array} \] \mathrm[/tex]


    which gives x_1 = 2 and x_2 = 1

    If I insert into the original equation, then values of x_1 and x_2 make it true.

    (b)

    Following the same method used in (a) I get x1 = -1 and x_2 = 3.

    (c)

    How do I use these results to prove that Span((1,3,2),(0,1,2)) = Span((2,7,6),(-1,0,4)) ???

    Can I claim the set of vectors are dependent, and therefore their spans equal each other?

    Best Regards
    Fred
     
    Last edited: Oct 8, 2006
  2. jcsd
  3. Oct 8, 2006 #2

    StatusX

    User Avatar
    Homework Helper

    The span of a set of vectors is the set of all (finite) linear combinations of vectors in the set. So a vector is in span((1,3,2),(0,1,2)) iff it is of the form a(1,3,2)+b(0,1,2) for some real numbers a and b, and similarly for span((2,7,6),(-1,0,4)), with vectors in this space having the form c(2,7,6)+d(-1,0,4). See if you can always get a pair (c,d) from (a,b) and vice versa.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Urgent: Linear Algebra Question(Please verify)
Loading...