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Basis, spans, nullspace

  1. Oct 1, 2012 #1
    I attached 2 problems.

    For problem #1. I want to make sure I'm on the right track, to find the span of Null(A), i need to put matrix A in RREF form. By doing so I get
    x1=-2t
    x2=-t
    x3=s
    x4=u (using u because I'm using t to denote transpose)
    where x1 to x4 is for each respective column. x3 and x4 turns out to be the free variables (columns 3 and 4).
    This gives the vectors s[0 0 1 0]^t and u[-2 -1 0 1]^t
    Thus the vectors that span Null(A) = [0 0 1 0]^t and [-2 -1 0 1]^t



    The second problem is for the 2nd matrix S in the attached file. The question asks, determine whether S is a basis for the vector space V.
    To do this, I think I need to take the determinant of the 3 vectors in S. This gives a determinant of "-2." This implies that the vectors are linearly indepdent and thus they form a basis for V=R^3.


    Did i do these 2 problems correctly?
     

    Attached Files:

  2. jcsd
  3. Oct 2, 2012 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You seem to be confused as to what A is doing. A maps R3 to R4. The null space of A is a subspace of R3, not R4 and so has nothing to do with x3 and x4, which are components in R4. Are you clear on the definition of "null space"?

    You can write a linear transformation as a matrix and put it in RREF form but it is not necessary. The definition of "null space" of A is the set of vectors that A maps to 0. So (s, t, u), in R3, will be in the null space of A if and only if -2t= 0, -t= 0, s= 0, u= 0. That's pretty clear isn't it? The null space of this a is just the 0 vector, (s, t, u)= (0, 0, 0).



    No, it doesn't.

    For the second problem, your A is, I think,
    [tex]\begin{bmatrix} 1 & -1 & 0 & 1 \\ 2 & -2 & 0 & 2 \\ 0 & 1 & 0 & 1\end{bmatrix}[/tex]
    which maps R4 to R3. Since the problem specifically says "find a set that spans the null space of A", and that is a subspace of R4, you need vectors in R4, not R3.

    You seem to be intent on using "formulas" rather than thinking about what these things mean. That's a bad idea!
     
    Last edited: Oct 2, 2012
  4. Oct 2, 2012 #3
    I didn't include the problem statement for the second problem. The "find vectors that span Nullspace(A)" is for matrix A. For S, then problem statement is "determine whether S is a basis for the vector space V." This was not included in the picture I attached. In order for S to be a basis for the vector space V, the vectors inside S must be linearly independent, thus det(S)≠0. In this case the determinant is not 0,thus it forms a basis for vector space V=R^3.
     
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