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Math Help - question about subspace

  1. #1
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    question about subspace

    I am new with vector space and subspace. The question asks "is the set of all vectors (x,y) where 0 \leq x \leq 1, 0 \leq y \leq 1 is in subspace  \mathbb{R}^2

    So there's 10 properties a "candidate" must satisfy to be classified as a subspace. These properties are listed alpha a-d beta e-h. I don't know if alpha and beta have some sort of special significance. In the book I'm reading there are two knew operators: vector addition and scalar multiplication. These confuse me because they appear to act in the same way as regular addition and multiplication.

    Here is my attempt at the question:
    \alpha)
    \mathbf{u}+\mathbf{v}=(a_1+a_2,b_1+b_2)
    this doesn't hold because (1,1)+(1,1)=(2,2) which is outside of the vector space
    a)
    \mathbf{u}+\mathbf{v}=(a_1,b_1)+(a_2,b_2)=(a_1+a_2  ,b_1+b_2)=(a_2+a_1,b_2+b_1)=\mathbf{v}+\mathbf{u} so this holds
    b)
    \mathbf{u}+(\mathbf{v}+\mathbf{w}) = (a_1,b_1)+[(a_2,b_2)+(a_3+b_3)]=(a_1+a_2,b_1+b_2)+(a_3,b_3)=(\mathbf{u}+\mathbf{v  })+\mathbf{w} this holds
    c)
    \mathbf{0}+\mathbf{u}=\mathbf{u} I guess this holds, I don't know how it could not.
    d)
    property: for each u in V, there is an element -u in V such that u -u = 0where denotes vector addition

    so I think this property does not hold true take for example u = 1. How do I say this in a more formal way?
    \beta)
    If u is any element of V and c is any real number then c scalar multiplication with u is in v.

    I said this does not hold if c is outside of [0,1]. Or is the idea that c has to be in [0,1] to start with?
    e)
    c \cdot (\mathbf{u}+\mathbf{v}=c \cdot [(a_1,b_2)+(a_2,b_2)]=c \cdot (a_1+a_2,b_1+b_2)
    I have a simillar problem, is c>1 or c<0 then it doesn't hold.
    f) g) and h) I have similar problem

    So since one property doesn't hold the answer to the question is no.

    I'm trying really hard here, can someone help me correct anymistakes I have?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by superdude View Post
    I am new with vector space and subspace. The question asks "is the set of all vectors (x,y) where 0 \leq x \leq 1, 0 \leq y \leq 1 is in subspace  \mathbb{R}^2

    So there's 10 properties a "candidate" must satisfy to be classified as a subspace. These properties are listed alpha a-d beta e-h. I don't know if alpha and beta have some sort of special significance. In the book I'm reading there are two knew operators: vector addition and scalar multiplication. These confuse me because they appear to act in the same way as regular addition and multiplication.

    Here is my attempt at the question:
    \alpha)
    \mathbf{u}+\mathbf{v}=(a_1+a_2,b_1+b_2)
    this doesn't hold because (1,1)+(1,1)=(2,2) which is outside of the vector space
    a)
    \mathbf{u}+\mathbf{v}=(a_1,b_1)+(a_2,b_2)=(a_1+a_2  ,b_1+b_2)=(a_2+a_1,b_2+b_1)=\mathbf{v}+\mathbf{u} so this holds
    b)
    \mathbf{u}+(\mathbf{v}+\mathbf{w}) = (a_1,b_1)+[(a_2,b_2)+(a_3+b_3)]=(a_1+a_2,b_1+b_2)+(a_3,b_3)=(\mathbf{u}+\mathbf{v  })+\mathbf{w} this holds
    c)
    \mathbf{0}+\mathbf{u}=\mathbf{u} I guess this holds, I don't know how it could not.
    d)
    property: for each u in V, there is an element -u in V such that u -u = 0where denotes vector addition

    so I think this property does not hold true take for example u = 1. How do I say this in a more formal way?
    \beta)
    If u is any element of V and c is any real number then c scalar multiplication with u is in v.

    I said this does not hold if c is outside of [0,1]. Or is the idea that c has to be in [0,1] to start with?
    e)
    c \cdot (\mathbf{u}+\mathbf{v}=c \cdot [(a_1,b_2)+(a_2,b_2)]=c \cdot (a_1+a_2,b_1+b_2)
    I have a simillar problem, is c>1 or c<0 then it doesn't hold.
    f) g) and h) I have similar problem

    So since one property doesn't hold the answer to the question is no.

    I'm trying really hard here, can someone help me correct anymistakes I have?
    If any of the properties don't hold then it isn't a subspace.
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  3. #3
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    in what case would there not be a zero vector?
    why is there the wired circle around the dot for scalar multiplication and the circle around the plus sign for vector addition?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by superdude View Post
    in what case would there not be a zero vector?
    why is there the wired circle around the dot for scalar multiplication and the circle around the plus sign for vector addition?
    What?
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  5. #5
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    how can there not be an element 0 in V such that u vector adition -u=0 ?

    what do these operators mean? how do they differ from normal addition and multiplication?
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by superdude View Post
    how can there not be an element 0 in V such that u vector adition -u=0 ?

    what do these operators mean? how do they differ from normal addition and multiplication?
    Oh, \odot, and \oplus.

    Give me an example, usually you consider V\oplus V' where they are both vector spaces.
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  7. #7
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    How about using the question I started with and showing how u and v from \mathbb{R}^2 such that u \oplus v and c \odot v is in \mathbb{R}^2

    so I'm confused as to what the differnce between + and \oplus and the \cdot and \odot is? Does the circle around them mean that the operations are being done to vectors?
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  8. #8
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by superdude View Post
    How about using the question I started with and showing how u and v from \mathbb{R}^2 such that u \oplus v and c \odot v is in \mathbb{R}^2

    so I'm confused as to what the differnce between + and \oplus and the \cdot and \odot is? Does the circle around them mean that the operations are being done to vectors?
    I've never seen that notation before, but I would assume it is what you said.
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  9. #9
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    ok thanks for saying that.
    My textbook is extremely unclear about what's going on:
    I was unsure if the definition was for real vector space, or if what is being defined is the operators themselves.

    I'm looking at page 272 of Introductory Linear Algebra An Applied First Course by Bernard Kolman and David R. Hill if anyone has this book and could clear up the meaning of \oplus , \odot that would be greatly appreciated.

    Another related question: it takes 10 requirments for something to be a vector space and only 2 of those 10 requirments for a set of matrices to be a subset of a vector space. Does that mean that to be a subset, the thing may not nescecarily be a vector space?
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