Question

Show that the following vectors form an inner product (in comments) ...

Answer 1

\[<u,v> = u_1v_1-u_2v_2-u_2v_1+3u_2v_2+2u_3v_3\] Where \[u=\left(\begin{matrix}u_1 \\ u_2\\u_3\end{matrix}\right), v=\left(\begin{matrix}v_1 \\ v_2\\v_3\end{matrix}\right)\]

Answer 2

this a real vector space right? and you are defining your inner product by the equation above

Answer 3

so you have to check 3 things :
\[<u,v>=<v,u>\] and you do it by writing it out and see if it works
that is
\[<u,v> = u_1v_1-u_2v_2-u_2v_1+3u_2v_2+2u_3v_3\]
\[<v,u>=v_1u_1-v_2u_2-v_2u_1+3v_2u_2+2v_3u_2\] and sure enough they are equal

Answer 4

you have to check that
\[<au,v>=a<u,v>\] which you can just about see with your eyeballs. you are going to factor out the a from the whole thing.
also that
\[<u+v,z>=<u,z>+<v,z>\]

Answer 5

and finally you have to show that
\[<u,u> \geq 0\]

Answer 6

the last one should say
\[u_1^2-u_2^2-u_2^2u_1+3u_2^2+3u_3^2\geq0\]

Answer 7

That was a great help! Thanks so much!