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Thread: Dimension of subspaces

  1. #1
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    Question Dimension of subspaces

    Hi,

    Just hoping that someone can help me better understand the dimension of a subspaces. To start, I will show this paragraph:

    Dimension of subspaces-screen-shot-2017-10-13-3.31.47-pm.png


    So there's a matrix that has 3 columns and 4 rows. I don't understand, fundamentally speaking, why the matrix would be a 2-dimensional subspace of R^4. I understand why it would be a subspace of R^4 (since each column has 4 components and therefore lies within the vector space of R^4), but I don't understand why it would be a 2-dimensional subspace. I know that the number of pivots indicates the dimension of the subspace (in this case there would be 2 pivots) - but I don't understand WHY this is the case. The mechanics of determining the subspace dimension is simple and what I want to understand is the theory behind the dimension of a subspace. I hope this makes sense.

    Please please help. I've been trying to wrap my head around this idea for hours :'(

    - otownsend
    Last edited by otownsend; Oct 13th 2017 at 12:50 PM.
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  2. #2
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    Re: Dimension of subspaces

    Because they're linearly dependent. The independent vectors produce a plane in R^4.
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  3. #3
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    Re: Dimension of subspaces

    Quote Originally Posted by otownsend View Post
    Hi,

    Just hoping that someone can help me better understand the dimension of a subspaces. To start, I will show this paragraph:

    Click image for larger version. 

Name:	Screen Shot 2017-10-13 at 3.31.47 PM.png 
Views:	12 
Size:	33.9 KB 
ID:	38195


    So there's a matrix that has 3 columns and 4 rows. I don't understand, fundamentally speaking, why the matrix would be a 2-dimensional subspace of R^4.
    Poor wording! A "matrix" is not a subspace. Your quote says that the column space of the matrix is a 2 dimensional subspace of R^4.

    I understand why it would be a subspace of R^4 (since each column has 4 components and therefore lies within the vector space of R^4), but I don't understand why it would be a 2-dimensional subspace. I know that the number of pivots indicates the dimension of the subspace (in this case there would be 2 pivots) - but I don't understand WHY this is the case. The mechanics of determining the subspace dimension is simple and what I want to understand is the theory behind the dimension of a subspace. I hope this makes sense.

    Please please help. I've been trying to wrap my head around this idea for hours :'(

    - otownsend
    The "column space" of a given matrix is the span of vectors formed by the columns of the matrix. Here, the matrix has four rows and three columns. The "four rows" mean that each column has four components so is in R^4. The three columns mean we are dealing with three vectors. Three vectors, if they were independent, would span a three dimensional subspace. But the fact that "the third column of the matrix is the sum of the first two columns" means that, writing v1, v2, and v3, as the first, second, and third columns, v3= v1+ v2 we can span the same subspace using just v1 and v2. That means the subspace is of dimension (at most) 2. I say "at most" because given that "the third column of the matrix is the sum of the first two columns" does not rule out that the first and third columns are both multiples of the first column- in which any vector in the column space is a multiple of the first column vector- and that would mean the column space has dimension 1.
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    Re: Dimension of subspaces

    Okay thank you for most part! That makes sense. In terms of it potentially being 1 dimension, couldn't the second column also be dependent on the third column? Why would we rule out that option?
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  5. #5
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    Re: Dimension of subspaces

    That could happen. The quote in your original post talks about a matrix, "A", as if it were given. Was there a matrix, "A", shown? If so, what is it?
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  6. #6
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    Re: Dimension of subspaces

    Alright I think I get what you mean. I unfortunately misplaced the document, I'll get back to you once I find it (not urgent however).
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  7. #7
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    Re: Dimension of subspaces

    The linear combinations you have don't span the whole of R^4. They just form a subspace of their own. That's all there is to it. Simple.
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