# Why is this function not an inner product space?

From Morris, A. O., Linear Algebra, an introduction (2nd edition, Van Nostrand, 1989) he gives the following as not being an inner product.

$(u,v)=x_1y_1-x_2y_1-x_1y_2+2x_2y_2$, where $u=(x_1,x_2),v=(y_1,y_2)$

My working below shows otherwise. Where do I go wrong?

Let $w=(z_1,z_2)$

$u+w=(x_1+z_1,x_2+z_2)$

$(u+w,v)=<(x_1+z_1,x_2+z_2),(y_1,y_2)>$

$=(x_1+z_1)y_1-(x_2+z_2)y_1-(x_1+z_1)y_2+2(x_2+z_2)y_2$

$=x_1y_1+z_1y_1-x_2y_1-z_2y_1-x_1y_2-z_1y_2+2x_2y_2+2z_2y_2$

$=x_1y_1-x_2y_1-x_1y_2+2x_2y_2+z_1y_1-z_2y_1-z_1y_2+2z_2y_2$

$=(u,v)+(w,v)$.

Similarly $(\alpha u,v)=\alpha(u,v)$

$(u,v)=(v,u)$ can easily be verified, and

$(u,u)=x_1y_1-x_2x_1-x_1x_2+2x_2x_2$

$=x_1^2-2x_1x_2+x_2^2+x_2^2$

$(x_1-x_2)^2+x_2^2 >0$ if $u \neq 0$.

• You should remove the leading spaces from your display equations in order for MathJax to render them properly. Also, you might find the \mathtt{align} environment helpful for aligning your equations in a particularly nice way. – Travis Willse May 5 '15 at 15:09
• Sorry I did not write the Latex code correctly for the subscripts and superscripts – Zilore Mumba May 5 '15 at 15:39

That product can be written as $$(u, v) = u^{t} A v$$ where $$A = \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}.$$ Since $A$ is symmetric, this is a perfectly good inner product as you've already shown. It's even positive-definite since $$(u,u) = x_1^2 - 2x_1 x_2 + 2x_2^2 = (x_1-x_2)^2 + x_2^2$$ is semipositive, and zero if and only if $x_1 = x_2 = 0$. Morris must have meant to write something else, but good for you to find the error!