- If $a \in \mathbb{R}$, let $\beta_a:\mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}$ be the billinear map defined by
$\beta_a \left( \left( \begin{array}{ccc} x_1 \\ x_2 \end{array} \right), \left( \begin{array}{ccc} y_1 \\ y_2 \end{array} \right) \right)=4x_1y_1-2x_1y_2-ax_2y_1+3x_2y_2$
Determine all $a$ such that $\beta_a$ defines an inner product on $\mathbb{R}^2$.
I'm really not sure hpw to move forward with this. The symmetry condition gives us that $2x_1y_2+ax_2y_1=2x_2y_1+ax_1y_2$ I think, simply by equating the given orientation to that when the vectors are 'switched round'.
I imagine we have to use linearity in the argument for it to be an IP, but I just can't see where. Help would be much appreciated.
The question (past exam question) goes on...
- Let $a \in \mathbb{R}$ s.t. $\beta = \beta_a$ is such an inner product. With respect to the inner product $\beta$, construct an orthonormal basis of $\mathbb{R}^2$ containing the vector $\left( \begin{array}{ccc} \frac{1}{2} \\ 0 \end{array} \right)$
Here, I'm assuming Gram-Schmidt needs to be applied, but I'm not entirely sure. I'll work at it and post any progress. Any assistance would be very appreciated.
Thanks in advance.