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 $\mathbb{R}^2$:

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

Actually, the four points that I have is (0,1),(1,0),(0,-1),(-1,0). Am I thinking in the right direction? — Sarah2018, Nov 01 '18 at 00:56
btw, it's better to include what you found in the question directly. — Siong Thye Goh, Nov 01 '18 at 01:05

score 2 · Answer 1 · answered Nov 01 '18 at 01:03

2

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.

answered Nov 01 '18 at 01:03

Siong Thye Goh

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

1 Answers1