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Linear transformation

  1. Mar 23, 2014 #1
    1. The problem statement, all variables and given/known data

    Say if f is a linear transformation from R2 to R3 with f(1,0) = (1,2,3) and f(0,1) = (0,-1,2).
    Determine f(x,y).


    3. The attempt at a solution


    I understand the theorem on linear transformation and bases but unsure as to how I should apply it in practice. Should I be performing the linear transformation test? But the question has already specified that f is a linear transformation.

    Edit: {u1,u2,u3...un} is a basis for Rn and {t1,t2,t3...tn} is a basis for Rm
    then there is a unique linear transformation such that f maps (u1) to t1: f(u1) = t1

    This can be expressed as f(u1) = t1, f(u2) = t2, f(u3) = t3...f(un) = tn

    f:R2 →R3

    f(e1) = (1,2,3)
    ∴f(1,0) = (1,2,3)
    ∴f(1) = 1, f(0) = 2

    f(e2) = (0,-1,2)
    ∴f(0,1) = (0,-1,2)
    ∴f(0) = 0, f(1) = -1
     
    Last edited: Mar 23, 2014
  2. jcsd
  3. Mar 23, 2014 #2

    HallsofIvy

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    A linear transformation, f, has the property that [itex]f(\alpha u+ \beta v)= \alpha fu+ \beta fv[/itex] where [itex]\alpha[/itex] and [itex]\beta[/itex] are scalars and u and v are vectors.

    You know f(0,1) and f(1, 0) so given any (x, y), apply f to x(1, 0)+ y(0, 1).
     
  4. Mar 23, 2014 #3

    pasmith

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    Homework Helper

    Can you write [itex](x,y) \in \mathbb{R}^2[/itex] as a linear combination of vectors for which you are given the value of [itex]f[/itex]?
     
  5. Mar 23, 2014 #4
    Hi halls, I've added some content to the OP.
     
  6. Mar 23, 2014 #5

    Suppose u = (e1) and v = (e2)
    f(u) = (1,2,3) and f(v) = 0,-1,2)

    f(λ1u + λ2v) = λ1f(u) + λ2f(v)
    = λ1 f(1,0) + λ2 f(0,1)
    ∴λ1(1,2,3) + λ2 (0,-1,2) = λ1(1,2,3) + λ2(0,-1,2)

    and if I am given λ1 and λ2 I can find a vector as a linear combination of
    λ1(1,2,3) + λ2(0,-1,2).
     
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