I have a simple question on which am seeking help clarifying. Am looking at two inner products, one which my text says is an inner product and I find it to be not, and another my text says it is but I find it to be not.
$$a_1b_1 + a_2b_1 + a_1b_2 + 2a_2b_2$$ which I put in matrix form as
$$(a_1,a_2) \left(\begin{matrix} 1 & 1\\ 1 & 2 \end{matrix}\right) \left(\begin{matrix} b_1 \\ b_2 \end{matrix}\right) $$ The matrix of coefficients is symmetric so I expect properties of inner products are satisfied. Next I have to show the matrix is positive definite. When I put it in upper triangular form
row2->row2-row1
I find the pivots are positive, $$ \left(\begin{matrix} 1 &-1 \\ 0 & 1 \end{matrix}\right) $$
hence the matrix is positive definite, hence the last property is satisfied but my textbook says it is not an inner product. The other function
$$a_1b_1+a_2b_1+a_1b_2-a_2b_2$$
$$(a_1,a_2) \left(\begin{matrix} 1 & 1\\ 1 & -1 \end{matrix}\right) \left(\begin{matrix} b_1 \\ b_2 \end{matrix}\right) $$ whose matrix I find to be negative definite
$$\left(\begin{matrix} 1 & 1\\ 0 & -2 \end{matrix}\right)$$
but my text says it is an inner product.