Results 1 to 6 of 6

Math Help - Which sets of vectors span R^3

  1. #1
    Banned
    Joined
    Feb 2009
    Posts
    97

    Which sets of vectors span R^3

    Which of the following sets of vectors span R^3?
    a. {(1,-1,2),(0,1,1)}
    b. {(1,2,-1),(6,3,0),(4,-1,2),(2,-5,4)}
    c. {(2,2,3),(-1,-2,1),(0,1,0)}
    d. {(1,0,0),(0,1,0),(0,0,1),(1,1,1)}
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,705
    Thanks
    1637
    Awards
    1
    Quote Originally Posted by antman View Post
    Which of the following sets of vectors span R^3?
    a. {(1,-1,2),(0,1,1)}
    b. {(1,2,-1),(6,3,0),(4,-1,2),(2,-5,4)}
    c. {(2,2,3),(-1,-2,1),(0,1,0)}
    d. {(1,0,0),(0,1,0),(0,0,1),(1,1,1)}
    What work have you done for yourself on any of these?

    I will tell you that the set in part a) does not. You tell us why?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Feb 2009
    Posts
    97
    Because it does not contain 3 vectors. I think I understand that the other 3 could be if 3 or more of the vectors are linearly independent, but I do not know how to find this.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Banned
    Joined
    Feb 2009
    Posts
    97
    For part a, my work is:

    c_{1}(1,-1,2)+c_{2}(0,1,1)=v=(a,b,c)

    c_{1}=a
    -c_{2}+c_{2}=b
    2c_{1}+c_{2}=c


     <br />
\begin{pmatrix}1 & 0 & a\\-1 & 1 & b\\2 & 1 & c\end{pmatrix}<br />

    Reduced
     <br />
\begin{pmatrix}1 & 0 & a\\0 & 1 & b+a\\0 & 0 & c-3a-b\end{pmatrix}<br />

    The system is inconsistent and has no solution so this set does not span R^3?

    Part b: same work resulting in the following reduced matrix

     <br />
\begin{pmatrix}1 & 0 & -2 & -{1/3}a+{2/3}b\\0 & 1 & 1 & {-1/9}b+{2/9}a\\0 & 0 & 0 & c-{1/3}a+{2/3}b\end{pmatrix}<br />
    So this would also not span R^3 since there are no solutions?

    Part c:

    c_{1}+c_{4}=a
    c_{2}+c_{4}=b
     <br />
c_{3}+c_{4}=c<br />
    So this set does span R^3?

    Does anyone know if this is the way this problem should be done? Thank you for your help.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,779
    Thanks
    1514
    Since two vectors can't span a three dimensional space, and you said that you understood that, I don't know why you did that.

    But that method can be used on the others: {(2,2,3),(-1,-2,1),(0,1,0)} spans R^3 if and only if there exist x, y, z such that x(2,2,3)+ y(-1,-2, 1)+ z(0, 1, 0)= (a, b, c) for any three numbers a, b, c. We can write that as 2x- y= a, 2x-2y+ z= b, 3x+ y= c. Adding the first and third equations, 5x= a+c or x= (a+c)/5. Then the third equation gives y= c- 3x= c- (3/5)a- (3/5)b. Put that into the second equation to see if you can solve it for z.

    Of course that is the same as reducing the augmented matrix
    \begin{bmatrix}2 & -1 & 0 & a \\2 & -2 & 1 & b \\3 & 1 & 0 & c\end{bmatrix}
    Actually, a, b, and c are irrelevant here. such x, y, z will exist for all a, b, c if and only if row reducing the matrix
    \begin{bmatrix}2 & -1 & 0 \\2 & -2 & 1 \\3 & 1 & 0\end{bmatrix}
    does not give a row consisting entirely of 0s. With four vectors you get a four rows and you only need three of them not consisting of 0s.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Banned
    Joined
    Feb 2009
    Posts
    97
    In my text, that is how the work is shown in the examples. When there was a set of 3 vectors for R^4, work was still shown in that format. So that's why I was so confused. But otherwise, my work is sufficient? Simply reduce and make sure there are enough nonzero rows. (ex. for a set of vectors to span R^3, there must be three nonzero rows after the matrix is reduced?)

    For part d, the matrix created was already reduced and all rows were nonzero (3 rows x 4 columns), so that would also span a 3-D space? Therefore, a and b are the only sets that would not. I just want to show enough work for my answers. Thank you!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Can 4 vectors in R^5 span R^4?
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 4th 2011, 08:40 PM
  2. span of vectors
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 28th 2009, 01:57 PM
  3. Vectors span R^3
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 21st 2009, 08:10 AM
  4. span of vectors
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: October 26th 2008, 09:47 AM
  5. Span vectors
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 25th 2008, 08:19 AM

Search Tags


/mathhelpforum @mathhelpforum