Given $ ||A||_p = \sup_{x \neq 0} \frac{||Ax||_p}{||x||_p}$ and
$$\begin{bmatrix} 1 & 2 \\ 0 & 2 \end{bmatrix}$$
Calculate $||A||_1$, $||A||_2$, and $||A||_\infty$.
Now I know how to calculate all these norms in practice, but what this question seems to be implying, is that I calculate the norms starting from the general definition of the induced matrix norms, and this is where I'm at a loss, is there a standard procedure to go about calculating them this way?
I'll do $\|A\|_1$ and you will do the rest.
Let $x=\binom{x_1}{x_2}\in \mathbb{R}^2$, by definition $\|x\|_1 = |x_1| + |x_2|$. Now $$Ax = \binom{x_1 + 2x_2}{2x_2}$$ and again by definition $\|Ax\|_1 = |x_1| + 2|x_2| +2|x_2| = |x_1|+4|x_2|$.
Therefore, $$\|A\|_1 = \sup_{x\not = 0} \frac{|x_1|+4|x_2|} {|x_1|+|x_2|}.$$
On one hand this is less or equal to $4$, because we can replace $|x_1|$ in the nominator by $4|x_1|$ and get a higher value. On the other hand if we take $x=(0,1)$ we get $\|Ax\|_1/\|x\|_1 = 4/1 = 4$. Therefore, $\|A\|_1=4$.