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Describing the intersection of two subspaces

  1. Oct 9, 2016 #1
    1. The problem statement, all variables and given/known data
    ##W_1 = \{(a_1, a_2, a_3) \in \mathbb{R}^3 : a_1 = 3a_3,~ a_3 = -a_2 \}##
    ##W_2 = \{(a_1, a_2, a_3) \in \mathbb{R}^3 : 2a_1 - 7a_2 + a_3 = 0 \}##
    Given that these are two subspaces of ##\mathbb{R}^3##, describe the intersection of the two, i.e. ##W_1 \cap W_2## and show that it is a subspace.

    2. Relevant equations


    3. The attempt at a solution

    Would it be sufficient just to say that ##W_1 \cap W_2 = \{(a_1, a_2, a_3) \in \mathbb{R}^3 : 2a_1 - 7a_2 + a_3 = 0,~a_1 = 3a_3,~ a_3 = -a_2 \}## and proceed to show that it is a subspace?
     
  2. jcsd
  3. Oct 9, 2016 #2

    FactChecker

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    You should look at those equations and you can draw some strong conclusions.
     
  4. Oct 9, 2016 #3
    Ah, so would it be more correct to say that ##W_1 \cap W_2 = \{ (0,0,0) \}##? If I leave it in the form ##W_1 \cap W_2 = \{(a_1, a_2, a_3) \in \mathbb{R}^3 : 2a_1 - 7a_2 + a_3 = 0,~a_1 = 3a_3,~ a_3 = -a_2 \}## is that incorrect?
     
  5. Oct 9, 2016 #4

    FactChecker

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    Yes. W1 is a line through (0,0,0) and W2 is a plane through (0,0,0). There are only two ways those can intersect. Either the W1 line is on the W2 plane or (0,0,0) is the only intersection point. In this case it is the later.
    It is technically correct but I assume that more is desired.
     
  6. Oct 10, 2016 #5
    So about in the case when we have
    ##W_1 = \{(a_1, a_2, a_3) \in \mathbb{R}^3 : a_1 = 3a_3,~ a_3 = -a_2 \}##
    ##W_2 = \{(a_1, a_2, a_3) \in \mathbb{R}^3 : a_1 - 4a_2 - a_3 = 0 \}##
    If we want to find ##W_1 \cap W_2##, would this just be ##W_1## since the intersection of a line and a three-dimensional space is just the line?
     
  7. Oct 10, 2016 #6

    Mark44

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    Your reasoning is wrong here. Neither of these subspaces is the entire R3. Both of them are planes, and both include the point (0, 0, 0).
     
  8. Oct 10, 2016 #7
    So in that case can I only leave ##W_1 \cap W_2## like ##\{(a_1, a_2, a_3) \in \mathbb{R}^3 : a_1 = 3a_3,~ a_3 = -a_2,~a_1 - 4a_2 + a_3 = 0 \}##
     
  9. Oct 10, 2016 #8

    Mark44

    Staff: Mentor

    It depends on what they're asking for, which could be what sort of geometric object is represented by this intersection, what's its equation, what is a basis for this intersection, among others.
     
  10. Oct 10, 2016 #9

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    I think you calculated that wrong. If
    ##W_2 = \{(a_1, a_2, a_3) \in \mathbb{R}^3 : a_1 + 4a_2 + a_3 = 0 \}##
    then I would agree that ##W_1 \cap W_2## = ##W_1## since ##3a_3 + 4(-a_3) + a_3 = 0## so all the points on W1 are on the plane W2.
     
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