(adsbygoogle = window.adsbygoogle || []).push({}); 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 allx[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.

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

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