## Elementary Linear Algebra 7th Edition

Let $u=(1,1,2), v=(0,1,0), w=(0,1,1)$. Now, using the definition of the function we have $$\langle u,w\rangle=\langle (1,1,2),(0,1,1)\rangle=3,$$ $$\langle v,w\rangle=\langle (0,1,0),(0,1,1)\rangle=0$$ $$\langle u+v,w\rangle=\langle (1,2,2),(0,1,1)\rangle=4,$$ One can see that $$\langle u+v,w\rangle\neq \langle u,w\rangle+\langle v,w\rangle .$$ Hence, the function does not define an inner product.