Question

The function <,>: R^2×R^2 to R :
< [X1], [Y1] > = X1^2+X2^2 define an inner
< [ X2], [Y2] >
Product over R^2.
True or false with full explanation

Answer 1

If $x=(x_1,x_2)$ and $y=(y_1,y_2)$

If $<x,y>= x_1^2+x_2^2$

We see that $<y,x>= y_1^2+y_2^2$

i.e $<x,y> \neq <y,x>$

Hence it does not define an inner product.