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I came across this question in Linear Transformations.

Find all Linear Maps $L\colon\mathbb{R}^3\longrightarrow\mathbb{R}^3$ whose kernel is exactly the plane $\{(x_1,x_2,x_3)∈\mathbb{R}^3\,|\,x_1+2x_2-x_3=0\}$.

How do I find the required Linear Transformations and how to denote them without going into specifying the corresponding matrix associated with the Transformation(I want to understand the Transformation first than the associated matrix)?

I do understand the terminology and I want to know the method to go about.

One of the answers I have seen directly gave me the matrix.

José Carlos Santos
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Yaksha
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1 Answers1

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Consider the basis $\{e_1,e_2,e_3\}$ of $\mathbb{R}^3$ such that $e_1=(1,0,1)$, $e_2=(0,1,2)$, and $e_3=(1,0,0)$. Note that $\{e_1,e_2\}$ is a basis of your plane. Therefore, given a linear map $f\colon\mathbb{R}^3\longrightarrow\mathbb{R}^3$, $\ker f$ is your plane if and only if $f(e_1)=f(e_2)=(0,0,0)$. On the other hand, $f(e_3)$ can be any vector $(a,b,c)$.

Now, note that $(0,1,0)=-2e_1+e_2+2e_3$ and that $(0,0,1)=e_1-e_3$. Therefore, if $(x,y,z)\in\mathbb{R}^3$, then\begin{align}f(x,y,z)&=f\bigl(x(1,0,0)+y(0,1,0)+z(0,0,1)\bigr)\\&=xf(1,0,0)+yf(0,1,0)+zf(0,0,1)\\&=x(a,b,c)+2y(a,b,c)-z(a,b,c)\\&=\bigl(a(x+2y-z),b(x+2y-z),c(x+2y-z)\bigr)\\&=(x+2y-z)(a,b,c).\end{align}

José Carlos Santos
  • 427,504
  • Could you please explain the steps in the solution in a more detailed way? What is the motivation behind choosing $e1,e2,e3$ in that way? Also, I did not understand how $e1$ & $e2$ form basis for the plane. Please forgive if these are simple ones to answer, but I am still learning... – Yaksha Oct 23 '17 at 10:52
  • @Yaksha Since $(1,0,1)$ and $(0,1,2)$ are linear independent vectors of your plane and since the plane is $2$-dimensional, they are a basis of the plane. Then I added a third vector (I chose $(1,0,0)$, more or less at random) outside the plane in order to get a basis of $\mathbb{R}^3$. – José Carlos Santos Oct 23 '17 at 10:56
  • Thank you. Now I understand it. In the third simplification step, shouldn't $yf(0,1,0)$ be replaced by $+2y(a,b,c)$ and $zf(0,0,1)$ be replaced by $-z(a,b,c)$? – Yaksha Oct 23 '17 at 11:31
  • @Yaksha Sure. My mistake. Is everything well now? – José Carlos Santos Oct 23 '17 at 11:35
  • Yeah! For future readers with doubts on basis of a plane, please refer this question https://math.stackexchange.com/q/251822 – Yaksha Oct 23 '17 at 11:54
  • So to effectively tell about the method, it's all about applying whatever conditions that are to be met, on the basis and then just taking scalar multiple to extend it to the entire space. Am I right? – Yaksha Oct 23 '17 at 12:08
  • @Yaksha Yes, that's how I did it. – José Carlos Santos Oct 23 '17 at 13:02