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Math Help - [SOLVED] is this the subspace?

  1. #1
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    [SOLVED] is this the subspace?

    7.

    v1 (1,0,1)
    v2 (0,1,1)
    v3 (-2,3,1)

    a) show all 3 vectors are linearly independent. I calculated the det and it is 0 so they are all linearly independent.

    I also found the zero vector, v3, which is v3 = 3*v2 - 2*v1

    b) describe, geometrically, the subspace of r3 spanned by v1, v2 and v3.

    i do not know how to do this. please help.

    c) find a vector w such that v1 and v2 and w are linearly independent.

    this one i'm not sure about, my teacher said i could just calculate the cross product.

    the cross product of v1 and v2 is (-1,-1,1). now what do i do with this? how do write the answer?
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  2. #2
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    Quote Originally Posted by thekrown View Post
    7.

    v1 (1,0,1)
    v2 (0,1,1)
    v3 (-2,3,1)

    a) show all 3 vectors are linearly independent. I calculated the det and it is 0 so they are all linearly independent.


    It's exactly the other way around : iff the determinant of the matrix formed by the vectors is zero are these vectors lin. dependent ...and they

    indeed are lin. dependent!



    I also found the zero vector, v3, which is v3 = 3*v2 - 2*v1

    b) describe, geometrically, the subspace of r3 spanned by v1, v2 and v3.

    i do not know how to do this. please help.


    It's just the plane spanned by v_1,v_2 .


    c) find a vector w such that v1 and v2 and w are linearly independent.

    this one i'm not sure about, my teacher said i could just calculate the cross product.


    Of course: the cross product of two vectors is a vector perpendicular to both vectors and, thus, lin. independent of them...


    the cross product of v1 and v2 is (-1,-1,1). now what do i do with this? how do write the answer?


    Just write that v_1,v_2,(-1,-1,1) are lin. ind., and check this showing their determinant is not zero!

    Tonio

    .
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  3. #3
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    Okay so to answer b I can simply write

    the subspace of r3 is the planned by v1 and v2 and this is a sufficient answer?
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  4. #4
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    Quote Originally Posted by thekrown View Post
    Okay so to answer b I can simply write

    the subspace of r3 is the planned by v1 and v2 and this is a sufficient answer?

    How can anyone not taking that class know!? Perhaps your teacher wants to write explicitly the plane, or for him/her it is sufficient the above answer...

    Tonio
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