$\langle u,v\rangle = [u]^T_B A_B[v]_B$ definition

Question

My textbooks says: An inner product $\langle u,v\rangle = [u]^T_B A_B[v]_B$ with bases $B = \langle b_1, b_2\rangle$

So does it work for all $u,v ∈ V$?

I don't see any proof or further explanation on the textbook, so how can I prove this definition?

Thanks a lot!

Answer 1

It is probably this:

Write $u = u_1 b_ 1 + u_2 b_ 2$ and $v = v_1 b_ 1 + v_2 b_ 2$.

Then, by bilinearity of $\langle \cdot , \cdot \rangle$, we have $$ \langle u,v \rangle = u_1 v_1 \langle b_1, b_1 \rangle + u_1 v_2 \langle b_1, b_2 \rangle + u_2 v_1 \langle b_2, b_1 \rangle + u_2 v_2 \langle b_2, b_2 \rangle $$ which can be written in matrix form as $$ \langle u,v \rangle = \pmatrix{ u_1 & u_2} \pmatrix{ \langle b_1, b_1 \rangle & \langle b_1, b_2 \rangle \\ \langle b_2, b_1 \rangle & \langle b_2, b_2 \rangle} \pmatrix{ v_1 \\ v_2} $$ The matrix is called the Gramian matrix of the given inner product.