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Range in Linear Transformation

  1. Apr 11, 2013 #1
    1. The problem statement, all variables and given/known data
    L: R^3 -> R^2
    What is the basis, and dim of the Range?

    2. Relevant equations

    3. The attempt at a solution
    So clearly L(x)= (0,0)^T. So the basis must be the empty space and dim is zero, right?

    Now, going of this same logic, Say L(x)=(x2,x3)^T. The basis would be {(1,0)^T, (0,1)^T} does this mean the range is just the Span of these two linearly independent vectors--> Span (v1,v2)?
  2. jcsd
  3. Apr 11, 2013 #2


    Staff: Mentor

    Every vector space must consist of at least the zero vector, so a basis for the range of L would be <0, 0>. The dimension of the range is zero, which is what you said.
    That's what a basis is - a set of vectors that spans some space or subspace. In this example, the range is all of R2.
  4. Apr 11, 2013 #3
    #44, for the win. Thanks you.
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