Chapter 5.1, Problem 19P

For Problems $19-21$, determine the inner product of the given vectors using (a) the inner product $\left(\mathrm{5.1.14}\right)$ in Problem $18$, (b) the standard inner product in ${ℝ}^2$.

$v=\left(1,0\right)$, $w=\left(-1,2\right)$.

18. Consider the vector space ${ℝ}^2$. Define the mapping $\langle ,\rangle$ by

$\langle v,w\rangle =2v_1{w}_1+v_1{w}_2+v_2{w}_1+2v_2{w}_2\left(\mathrm{5.1.14}\right)$

for all vectors $v=\left(v_1,v_2\right)$ and $w=\left(w_1,w_2\right)$ in ${ℝ}^2$. Verify that Equation $\left(\mathrm{5.1.14}\right)$ defines an inner product on ${ℝ}^2$.