Proof of a norm that is a Matrix Vector multiplication.

$$\begin{array}{l}{\text { We have following matrix }} \\ {\qquad T=\left(\begin{array}{cc}{-3} & {1} \\ {1} & {2}\end{array}\right)} \\ {\text { a) Show that, with }\|x\|_{T}=\|T x\|_{\infty} \text { a vector norm is defined for } \mathbb{R}^{2} \text { }} \\ {\text { (Hint: Properties of a norm.) }}\end{array}$$ $$\begin{array}{l}{\text { b) Sketch the unit circle } B_{T}=\left\{x \in \mathbb{R}^{2}\|x\|_{T} \leq 1\right\}} \\ {\text { c) State the }\|\cdot\| T \text { assigned matrix norm }} \\ {\quad\|A\|_{T}:=\max _{x \neq 0} \frac{\|A x\|_{T}}{\|x\|_{T}}} \\ {\text { with the matrix norm}\|\cdot\|_{\infty} \text { }}\end{array}$$

Am trying to do a. So if the vector norm is 0 the inside has to be 0. Is the vector norm the square root and the inside ^2 ? I literally have no clue how to proof the properties.. Thanks for any help.

• Hint: The triangle inequality follows easily by the definition given. What is the T-norm of $x+y$? – Sean Roberson Nov 4 '19 at 8:08

They are defining a new norm in $$\mathbb{R}^2$$ by the expression $$\|x\|_T=\|Tx\|_{\infty}.$$ Take into account that if $$x=(a,b)$$ then $$\|(a,b)\|_{\infty}=\max\{|a|,|b|\}$$ in your particular case $$\|x\|_T=\|Tx\|_{\infty}=\max\{|-3a+b|,|a+2b|\}.$$
• $$\begin{array}{l}{\|x\|=0 \Rightarrow x=0} \\ {\|\alpha \cdot x\|=|\alpha| \cdot\|x\|} \\ {\|x+y\| \leq\|x\|+\|y\|}\end{array}$$ now i'm trying to proof it is a norm. Did it this way $$\mid \mid Tx \mid \mid_{\inf} A := Tx. \mid \mid aA \mid \mid = max \{ \mid a \mid ( \mid \mid -3a+b \mid ), \mid a \mid ( \mid a+2b \mid ) = \mid a \mid \mid \mid A \mid \mid$$ – Rack Cloud Nov 4 '19 at 9:53
• $$\mid \mid A+y \mid \mid \leq \mid \mid A \mid \mid + \mid \mid y \mid \mid$$ $$\mid \mid A+y \mid \mid = max\{ \mid -3a+b+y_{1} \mid, \mid a+2b+y_{2} \}$$ $$\mid \mid A \mid \mid + \mid \mid y \mid \mid = max\{ \mid -3a+b\mid, \mid a+2b \mid \}+ max \{ \mid y_{1}\mid, \mid y_{2}\mid \}$$ Don't know how to go to the equation with $$\leq$$ – Rack Cloud Nov 4 '19 at 9:59
• Ok guess i will state it like that can anyone help me with b) ? How do i sketch $$\max \left\{\left|-3 a+b+y_{1}\right|, | a+2 b+y_{2}\right\}$$ – Rack Cloud Nov 4 '19 at 15:47