1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook