Find the four points $b \in \mathbb{R}^2$ such that $||b||_2=1$ and $||Ab||_2=1$

Question

Define a matrix $A=\begin{bmatrix} 3 & -1 \\1 & 0 \end{bmatrix}$ which represents a linear transformation in $\mathbb{R}^2$. Consider the following norms on $\mathbb{R}^2$:

$||(x,y)||_p=(x^p+y^p)^{1/p}\\ ||(x,y)||_\infty=max{|x|,|y|}$

Find the four points $b \in \mathbb{R}^2$ such that $||b||_2=1$ and $||Ab||_2=1$

Answer 1

We are trying to solve for $$b_1^2+b_2^2=1$$

$$(3b_1-b_2)^2+b_1^2=1$$

Note that this means $$(3b_1-b_2)^2=b_2^2$$

and hence $$(3b_1-b_2-b_2)(3b_1-b_2+b_2)=0$$

Hopefully you can take it from here.