How do I solve for $\langle w_1,w_2\rangle $ when the vectors are given relative to these particular basis?

Question

If I have $\vec{w}_1=(2,3)$ and $\vec{w}_2=(1,1)$, but they are relative to the basis $\vec{u}=(1,1), \vec{v}=(1,-1)$. How do I find the scalar product of $w_1$ and $w_2$?

I know that $\langle w_1,w_2\rangle =2\cdot 1+3\cdot 1$ when the basis are orthogonal, but that is not the case here. Would I say that $w_1=2\cdot 1+3\cdot 1$ and $w_2=1\cdot 1 + 1\cdot -1$? If so, then how do I proceed from here?

Answer 1

We have that in the standard basis

$$\langle w_1,w_2\rangle=w_1^Tw_2$$

and indicating with $M$ the matrix for the change of basis from the the new basis $\mathcal{B}$ to the standard basis we have

$$\langle w_1,w_2\rangle=(Mw_{1,\mathcal{B}})^T\, Mw_{2,\mathcal{B}}=w_{1,\mathcal{B}}^TM^T Mw_{2,\mathcal{B}}$$

therefore the key point is to find the matrix $M$.

Answer 2

Use bilinearity of the scalar product: You’re given that $\vec w_1 = a_1\vec u+b_1\vec v$ and $\vec w_2 = a_2\vec u+b_2\vec v$. Expand $$\langle \vec w_1,\vec w_2\rangle = \langle a_1\vec u+b_1\vec v, a_2\vec u+b_2\vec v\rangle$$ into a sum of terms involving scalar products of $\vec u$ and $\vec v$.