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Math Help - Subspaces & spans

  1. #1
    Junior Member SirOJ's Avatar
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    Subspaces & spans

    Determine (giving reasons for your answer) whether or
    not the following subsets of
    R3 are subspaces of R3:
    (i)
    {(x, y, z) | xy = 0}

    (ii) {(x, y, z) | x = 3z}

    for (i) I got no, the set is not a subset of R3
    for (ii) Yes, the set is a subset of R3

    Could someone please tell me if I'm right or not...?

    What is the
    span of a set of vectors in Rn? Does the set of vectors

    {(1, 2, 4), (2, 1, 3), (4,1, 1)} span R3? Give reasons for your answer.

    Would someone be able to step through this problem for me please...?
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  2. #2
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    Your answers are correct for (i) and (ii).

    A set of three vectors spans \mathbb{R}^3 if and only if they are linearly independent over \mathbb{R}. Do you know how to find out if a set of vectors is linearly independent?
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  3. #3
    Junior Member SirOJ's Avatar
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    yes i think so... I just want to see if the set spans R3 not if it's linearly independent. I thought I would only need to check if it's linearly independent if I was lookin for a basis of R3

    i Know that you can write any element in R3 as a combonation of S (where S is equal to the set)

    e.g. (x,y,z) = a1(1,2,4) + a2(2,1,3) + a3(4,-1,1)

    => x = a1 + 2a2 + 4a3
    y = 2a1 + a2 - a3
    z = 4a1 + 3a2 + a3

    It's at this stage I get stuck... I think I am ment to write a1, a2, a3 in terms of x, y, z...?

    I'm not sure if im able to do this or if this is even the right method

    Any help would be much appreciated
    Last edited by SirOJ; December 3rd 2009 at 12:59 PM. Reason: submitted reply without realising
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  4. #4
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    Quote Originally Posted by Defunkt View Post
    Your answers are correct for (i) and (ii).

    A set of three vectors spans \mathbb{R}^3 if and only if they are linearly independent over \mathbb{R}. Do you know how to find out if a set of vectors is linearly independent?
    for every \vec{x}\in \mathbb{R}^{3} there's \left (\alpha_{1},\alpha _{2},\alpha _{3}  \right )\in \mathbb{R}^{3} such that
    \vec{x}=\alpha_{1}(1,2,4) + \alpha _{2}(2,1,3) + \alpha _{3}(4,-1,1)
    no ?
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  5. #5
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    Quote Originally Posted by SirOJ View Post
    yes i think so... I just want to see if the set spans R3 not if it's linearly independent. I thought I would only need to check if it's linearly independent if I was lookin for a basis of R3

    i Know that you can write any element in R3 as a combonation of S (where S is equal to the set)

    e.g. (x,y,z) = a1(1,2,4) + a2(2,1,3) + a3(4,-1,1)

    => x = a1 + 2a2 + 4a3
    y = 2a1 + a2 - a3
    z = 4a1 + 3a2 + a3

    It's at this stage I get stuck... I think I am ment to write a1, a2, a3 in terms of x, y, z...?

    I'm not sure if im able to do this or if this is even the right method

    Any help would be much appreciated

    Use Gauss elemination method.
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  6. #6
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    Quote Originally Posted by Raoh View Post
    for every \vec{x}\in \mathbb{R}^{3} there's \left (\alpha_{1},\alpha _{2},\alpha _{3}  \right )\in \mathbb{R}^{3} such that
    \vec{x}=\alpha_{1}(1,2,4) + \alpha _{2}(2,1,3) + \alpha _{3}(4,-1,1)
    no ?
    That is incorrect.

    Definiton: Let V be a vector space over a field \mathbb{F} and x_1,x_2,...,x_n \in V. We say that x_1,x_2,...,x_n are linearly dependent if there exist \alpha_1,\alpha_2,...,\alpha_n \in \mathbb{F}, not all zero, such that:

    \sum_{i=1}^{n}\alpha_ix_i = 0.

    We will say that y_1,...,y_n are linearly independent if \sum_{j=1}^{n}b_jy_j = 0 \Leftrightarrow ~ b_1 = b_2 = ... = b_n = 0
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  7. #7
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    Quote Originally Posted by Defunkt View Post
    That is incorrect.

    Definiton: Let V be a vector space over a field \mathbb{F} and x_1,x_2,...,x_n \in V. We say that x_1,x_2,...,x_n are linearly dependent if there exist \alpha_1,\alpha_2,...,\alpha_n \in \mathbb{F}, not all zero, such that:

    \sum_{i=1}^{n}\alpha_ix_i = 0.

    We will say that y_1,...,y_n are linearly independent if \sum_{j=1}^{n}b_jy_j = 0 \Leftrightarrow ~ b_1 = b_2 = ... = b_n = 0
    i know the definition ,i meant that a set spans R^3 if every vector in R^3 can be written as a linear combination of the vectors in that set.
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  8. #8
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    Quote Originally Posted by Raoh View Post
    i know the definition ,i meant that a set spans R^3 if every vector in R^3 can be written as a linear combination of the vectors in that set.
    Oops.

    For some reason I confused the spanning set with a base. My bad. The correct definition for a spanning set is what Raoh had written in his previous post.
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  9. #9
    Junior Member SirOJ's Avatar
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    Thanks a lot guys, i'll give gaussian elimination a try...
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  10. #10
    Junior Member SirOJ's Avatar
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    Quote Originally Posted by Raoh View Post

    Use Gauss elemination method.

    x = a1 + 2a2 + 4a3
    y = 2a1 + a2 - a3
    z = 4a1 + 3a2 + a3

    x = a1 + 2a2 + 4a3
    y-2x = 0 - 3a2 - 9a3 R2-> R2 - 2R1
    z-2y = 0 + a2 + 3a3 R3 -> R3 - 2R2

    x = a1 + 2a2 + 4a3
    y-2x = 0 - 3a2 - 9a3
    (z-2y) + 1/3(y-2x) = 0 R3-> R3 + 1/3R2


    Any ideas were to go from here...? I'm sorry for all tge questions, it's just it's really frustrating getting stuck mid-problem
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  11. #11
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    Quote Originally Posted by SirOJ View Post
    x = a1 + 2a2 + 4a3
    y = 2a1 + a2 - a3
    z = 4a1 + 3a2 + a3

    x = a1 + 2a2 + 4a3
    y-2x = 0 - 3a2 - 9a3 R2-> R2 - 2R1
    z-2y = 0 + a2 + 3a3 R3 -> R3 - 2R2

    x = a1 + 2a2 + 4a3
    y-2x = 0 - 3a2 - 9a3
    (z-2y) + 1/3(y-2x) = 0 R3-> R3 + 1/3R2


    Any ideas were to go from here...? I'm sorry for all tge questions, it's just it's really frustrating getting stuck mid-problem
    you should cancel the last line with the first one.
    l_{3}=l'_{3}-4l_{1}
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  12. #12
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    It is simpler to throw out the "a"s and just do the Gaussian elimination on the matrix of coefficients:
    \begin{bmatrix} 1 & 2 & 4 \\ 2 & 1 & 3 \\ 4 & -1 & 1\end{bmatrix}
    Subtract twice the first row from the second row and subtract 4 times the first row from the third row:
    \begin{bmatrix}1 & 2 & 4 \\ 0 & -3 & -5 \\ 0 & -9 & -15\end{bmatrix}
    Subtract 3 times the second row from the third
    \begin{bmatrix}1 & 2 & 4 \\ 0 & -3 & -6 \\ 0 & 0 & 0\end{bmatrix}.

    There you have it: Since the last row is all 0s those vectors are not independent. Since no two are multiples of each other, take any two of the three matrices as basis.
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  13. #13
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    Quote Originally Posted by HallsofIvy View Post
    It is simpler to throw out the "a"s and just do the Gaussian elimination on the matrix of coefficients:
    \begin{bmatrix} 1 & 2 & 4 \\ 2 & 1 & 3 \\ 4 & -1 & 1\end{bmatrix}
    Subtract twice the first row from the second row and subtract 4 times the first row from the third row:
    \begin{bmatrix}1 & 2 & 4 \\ 0 & -3 & -5 \\ 0 & -9 & -15\end{bmatrix}
    Subtract 3 times the second row from the third
    \begin{bmatrix}1 & 2 & 4 \\ 0 & -3 & -6 \\ 0 & 0 & 0\end{bmatrix}.

    There you have it: Since the last row is all 0s those vectors are not independent. Since no two are multiples of each other, take any two of the three matrices as basis.
    i think he's trying to prove if that set of vectors spans \mathbb{R}^3.
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  14. #14
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    start like this
    \left\{\begin{matrix}<br />
a_{1} + 2a_{2} + 4a_{3}=x  \\ <br />
2a_{1} + a_{2} - a_{3}=y \\ <br />
4a_{1}+ 3a_{2} + a_{3}=z<br />
\end{matrix}\right.
    use Gauss method well and you should end up with this equation(if i'm not mistaking),
    \frac{2}{3}x+\frac{5}{3}y-z=0
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  15. #15
    Junior Member SirOJ's Avatar
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    Quote Originally Posted by Raoh View Post
    start like this
    \left\{\begin{matrix}<br />
a_{1} + 2a_{2} + 4a_{3}=x \\ <br />
2a_{1} + a_{2} - a_{3}=y \\ <br />
4a_{1}+ 3a_{2} + a_{3}=z<br />
\end{matrix}\right.
    use Gauss method well and you should end up with this equation(if i'm not mistaking),
    \frac{2}{3}x+\frac{5}{3}y-z=0
    Yes I am only interested to see if the set spans R3

    By the getting the above equation what does that prove or what is the next step...?

    p.s. i'm not 100% sure how you got that equation (and I do know how to use gaussian elimination)
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