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Finding a basis for V

  1. Mar 23, 2009 #1
    1. The problem statement, all variables and given/known data
    Let V be the vector space spanned by v1 = cos^2(x) , v2 = sin^2(x) , v3 = cos(2x) .
    Show that
    {v1 ,v2 ,v3} is not a basis for V , then find a basis for V .
    2. Relevant equations



    3. The attempt at a solution
    (-1)*cos^2(x) + (1)*sin^2(x) + (1)*cos(2x)=0
    {v1 ,v2 ,v3} is not linearly independent, so is not a basis for V.

    I am not sure how to do the next part of the question, "find a basis for V" .
    I am thinking its probably {v1,v2}. As v1 and v2 are linearly independent. However how do I show this set spans V?
     
  2. jcsd
  3. Mar 23, 2009 #2

    Mark44

    Staff: Mentor

    V is the space spanned by v1 = cos^2(x), v2 = sin^2(x), and v3 = cos(2x). Since cos(2x) is a linear combination of v1 and v2, we can remove v3 without removing any of the vectors in V.

    What does it mean (i.e., the definition) to say a space is spanned by a set of vectors?
     
  4. Mar 23, 2009 #3
    To span a space means that every vector in the space can be written as a linear combination in the set.
     
  5. Mar 23, 2009 #4
    Write v (an arbitrary vector in V) as a linear combination of v1, v2, and v3 and then see if you can write v as a linear combination of just v1 and v2. Hint: v3 = v1-v2.
     
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