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Linear algebra: Find the span of a set

  1. Dec 22, 2015 #1
    1. The problem statement, all variables and given/known data
    Find the span of [itex]U=\{2,\cos x,\sin x:x\in\mathbb{R}\}[/itex] ([itex]U[/itex] is the subset of a space of real functions) and [itex]V=\{(a,b,b,...,b),(b,a,b,...,b),...,(b,b,b,...,a): a,b\in \mathbb{R},V\subset \mathbb{R^n},n\in\mathbb{N}\}[/itex]

    2. Relevant equations
    -Vector space span
    -Linear independence
    -Rank
    3. The attempt at a solution
    Objects in [itex]U[/itex] :[itex]2,\cos x,\sin x[/itex] are linearly independent, so they span [itex]\mathbb{R^3}[/itex].
    Let ,[itex]n=3\Rightarrow [V]= \begin{bmatrix}
    a & b & b \\
    b & a & b \\
    b & b & a \\
    \end{bmatrix}[/itex]

    [itex]rref[V]=\begin{bmatrix}
    1 & 0 & 0 \\
    0 & 1 & 0 \\
    0 & 0 & 1 \\
    \end{bmatrix}\Rightarrow [/itex] vectors in [itex]V[/itex] span [itex]\mathbb{R^3}[/itex], if [itex]a,b\neq 0[/itex].

    But because [itex]V\subset\mathbb{R^n}\Rightarrow [/itex] vectors span [itex]\mathbb{R^{n-1}}[/itex].

    Is this correct?
     
  2. jcsd
  3. Dec 22, 2015 #2

    Ray Vickson

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    No. Having ##a,b \neq 0## is not enough. Look at the determinant of your ##3 \times 3## matrix.
     
  4. Dec 22, 2015 #3
    Thanks, conditions [itex]a\neq b,a\neq -2b,a\neq 0,b\neq 0[/itex] must be satisfied.
    But are the span of [itex]U,V[/itex] correct?
    Also, what happens if the previous conditions aren't satisfied?
     
  5. Dec 22, 2015 #4

    Ray Vickson

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    That is a good question for you to think about. It might make a difference whether you have ##a = b## or ##a = -2b##.
     
  6. Dec 23, 2015 #5

    WWGD

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    But I think the question is on the span of ##U## , and , in your post you have that U is a subset of a function space. Function spaces are usually infinite-dimensional; I cannot think of any function space that is not.
     
  7. Dec 23, 2015 #6

    vela

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    What's the definition of span? The functions 2, cosine, and sine aren't elements of ##\mathbb{R}^3##, so how can they span ##\mathbb{R}^3##?

    No, it's not correct. In the ##n=3## case, are you claiming the three vectors span both ##\mathbb{R}^2## and ##\mathbb{R}^3##? How can that possibly work? The elements of ##\mathbb{R}^2## are ordered pairs while the elements of ##\mathbb{R}^3## are 3-tuples.
     
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