1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Show that T preserves scalar multiplication - Linear Transformations

  1. Oct 5, 2011 #1
    1. The problem statement, all variables and given/known data

    Let T:ℝ[itex]^{2}[/itex]→ℝ be defined by
    T[tex]\left(\begin{array}{c} x_{1} \\x_{2}\end{array}\right)[/tex] = (0 if x[itex]_{2}[/itex] = 0. [itex]\frac{x^{3}_{1}}{x^{2}_{2}}[/itex] otherwise.)

    Show that T preserves scalar multiplication, i.e T(λx) = λT(x) for all λ [itex]\in[/itex] ℝ and all x [itex]\in[/itex] ℝ[itex]^{2}[/itex]
    3. The attempt at a solution

    T(λx) = T[tex]\left(\begin{array}{c} (λx_{1}) \\(λx_{2})\end{array}\right)[/tex] = (λ0 = 0 if x[itex]_{2}[/itex] = 0, or [itex]\frac{(λx_{1})^{3}}{(λx_{2})^{2}}[/itex])
    = λT[tex]\left(\begin{array}{c} x_{1} \\x_{2}\end{array}\right)[/tex] = λ0 = 0 if x[itex]_{2}[/itex] = 0, or
    λ*[tex]\left(\begin{array}{c} (x_{1})^{3} \\(x_{2})^{2}\end{array}\right)[/tex]

    Is that a correct proof?

    It's a bit hard to read because whenever I try to put a vector, it puts it into a new line.
  2. jcsd
  3. Oct 5, 2011 #2


    User Avatar
    Science Advisor

    It's putting new lines wherever you try to put a vector because you are putting "bits and pieces" inside [ tex ] tags. Don't do that. Put entire equations inside [ tex ] tags.

    Also do "[itex]x_2= 0[/itex]" and "[itex]x_2\ne 0[/itex]" separately. You have "[itex]x_2= 0[/itex]" in two different places which makes it hard to read.

    If [itex]x_2= 0[/itex], then [itex]\lambda x_2= 0[/itex] for any [itex]\lambda[/itex] so [itex]T(\lambda x)= 0= \lambda T(x)[/itex].

    If [itex]x_2\ne 0[/itex], then [itex]\lambda x_2\ne 0[/itex] for any [itex]\lambda[/itex] except [itex]\lambda= 0[/itex] so you should do two separate cases, [itex]\lambda= 0[/itex] and [itex]\lambda\ne 0[/itex], here.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook