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Homework Help: Linear algebra problem related to vector subspace

  1. Apr 24, 2015 #1
    1. The problem statement, all variables and given/known data
    X ={(x1,x2,x2 −x1,3x2):x1,x2 ∈R}
    f(x1,x2,x2 −x1,3x2)=(x1,x1,0,3x1)
    1. Find a basis for X.
    2. Find dim X.
    3. Find ker f and im f
    4. Find bases for ker f and im f
    5. Is f a bijection? Why?
    6. Find a diagonal matrix for f.

    2. Relevant equations

    3. The attempt at a solution
    1. Put x1=x2=1: (1, 1, 0, 3) and Put x1=1 and x2=2: (1, 2, 1, 6)
    2. Dim X = 2 as there are two vectors
    3. Ker f = 0, im f = X
    4. (0,0,0,0)
    5. I guess no, but do not know how to explain
    6. No idea
  2. jcsd
  3. Apr 24, 2015 #2


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    Staff Emeritus
    Science Advisor
    Gold Member

    1. Looks like a good guess, but you will need to prove that the two vectors you found span X.

    2. Here you will also need to prove that the two vectors you found in 1 are linearly independent.

    3. ker f is a set, not a vector or a number. If you meant ker f={0}, where 0 is the zero vector, then the answer is wrong since e.g. f(0,1,1,3)=0. Your answer for im f is wrong too, since (1,2,1,6) is in X but not in im f.
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