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Math Help - Are these subspaces of R^n?

  1. #1
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    Are these subspaces of R^n?

    Which ones of the following subsets of \mathbb{R}^n are subspaces.

    a) L=\{x= \[\alpha_1,\ldots,\alpha_n\]^\top\  | \sum_{i=1}^n (-1)^n\alpha_i =0\}

    b) M=\{x= \[\alpha_1,\ldots,\alpha_n\]^\top\  | \alpha_i - \alpha_{i-1} = constant\\ i= 2,3,\dots,n\}

    c) N=\{x= \[\alpha_1,\ldots,\alpha_n\]^\top\  | \left\frac{\alpha_i}{\alpha_{i-1}} \right = constant\\ i= 2,3,\dots,n\}

    d) P=\{x= \[\alpha_1,\ldots,\alpha_n\]^\top\  | \sum_{i=1}^n 2^{n-1}\alpha_i =0\}

    I should decide their dimensions and give a basis for each. I think a) and d) are subspaces, how can I find the dimension of these spaces and how can I give a basis?

    Thank you!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Are these subspaces of R^n?

    Quote Originally Posted by gotmejerry View Post
    Which ones of the following subsets of \mathbb{R}^n are subspaces. a) L=\{x= \[\alpha_1,\ldots,\alpha_n\]^\top\  | \sum_{i=1}^n (-1)^n\alpha_i =0\}
    (i) Obviously (0,\ldots,0)^T\in L (ii) Consider \alpha=(\alpha_1,\ldots,\alpha_n)^T\in L and \beta=(\beta_1,\ldots,\beta_n)^T\in L Then, \sum_{i=1}^n(-1)^n(\alpha_i+\beta_i)=\sum_{i=1}^n(-1)^n\alpha_i+\sum_{i=1}^n(-1)^n\beta_i=0+0=0 , so \alpha+\beta\in L (iii) ....

    What have you tried for the rest?.
    Last edited by FernandoRevilla; November 12th 2011 at 02:49 PM.
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  3. #3
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    Re: Are these subspaces of R^n?

    I did it this way. So when the linear combination of 2 random elements of the set gives an element which in the set too, it is a subspace.

    For c) the 0-vector isn't in the set so it cannot be a subspace.

    For d) I did the same what you did with a). It is a subspace.

    For b) I got from the lin.comb of 2 elements (a+b)*c when c is the constant and a,b are scalars, so I think I should have got c, so it is not a subspace.

    But my main problem is I cannot decide their dimensions and cannot give a basis
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: Are these subspaces of R^n?

    Quote Originally Posted by gotmejerry View Post
    But my main problem is I cannot decide their dimensions and cannot give a basis
    For example L\equiv \alpha_1+\ldots +\alpha_n=0 or equivalently:

    \begin{bmatrix} \alpha_1\\{\alpha_2}\\ \vdots\\{\alpha_n}\end{bmatrix}=\begin{bmatrix} \alpha_1\\{\alpha_2}\\ \vdots\\{-\alpha_1-\ldots -\alpha_{n-1}}\end{bmatrix}=\alpha_1\begin{bmatrix} 1\\{0}\\ \vdots\\{-1}\end{bmatrix}+\alpha_2\begin{bmatrix} 0\\{1}\\ \vdots\\{-1}\end{bmatrix}+\ldots

    Conclude.
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  5. #5
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    Re: Are these subspaces of R^n?

    My conclusion, its dimension is n-1. What about the d)?
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Re: Are these subspaces of R^n?

    Quote Originally Posted by gotmejerry View Post
    My conclusion, its dimension is n-1.
    Right.

    What about the d)?
    You wrote:

    Quote Originally Posted by gotmejerry View Post
    d) P=\{x= \[\alpha_1,\ldots,\alpha_n\]^\top\  | \sum_{i=1}^n 2^{n-1}\alpha_i =0\}
    In such case , P=L unless you meant \sum_{i=1}^n 2^{i-1}\alpha_i =0
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  7. #7
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    Re: Are these subspaces of R^n?

    And are my assumptions correct that b) and c) are not subspaces?
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  8. #8
    MHF Contributor FernandoRevilla's Avatar
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    Re: Are these subspaces of R^n?

    Quote Originally Posted by gotmejerry View Post
    And are my assumptions correct that b) and c) are not subspaces?
    M is subspace of \mathbb{R}^n and:

    \begin{bmatrix}\alpha_1\\{\alpha_2}\\ \vdots\\{\alpha_n}\end{bmatrix}=\begin{bmatrix} \alpha_1\\{\alpha_1+k}\\ \vdots\\{\alpha_1+(n-1)k}\end{bmatrix}=\alpha_1\begin{bmatrix}1\\{1}\\ \vdots\\{1}\end{bmatrix}+k\begin{bmatrix}0\\{1}\\ \vdots\\{n-1}\end{bmatrix}

    N is not a subspace of \mathbb{R}^n .
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