Scalar Product in Euclidean Spaces

What is the condition for a symmetric bilinear form to be a scalar product on Rn?

Which of the following properties does a scalar product on E satisfy?

What is the real number denoted by < x, y > in a scalar product on E?

What type of form is the canonical scalar product on Rn?

What is the necessary condition for a symmetric bilinear form on R2 to be a scalar product?

Definition 6.1.1: We call scalar product (or inner product) on E every definite positive symmetric bilinear form on E, i.e., every mapping f : E⇥E 7 . R which satisfies the following properties: i) For all x, x0 , y 2 E and a, a0 2 R, f (ax + a0 x0 , y) = af (x, y) + a0 f (x0 , y). ii) f (x, y) = f (y, x) for all x, y 2 E. iii) f (x, x) 0 for all x 2 E. iv) f (x, x) = 0 if and only if x = ______.

If E is equipped with a scalar product, then we say that E is an euclidean ______.

Remark 6.1.1: A scalar product on E is nondegenerate, and therefore it is of ______ n. since itisdefinite

By the proposition 5.6.3, a symmetric bilinear form f on E is a scalar product on E if and only if sg(f ) = (n, ______). positive definite

If f is a scalar product on E, then, for all (x, y) 2 E ⇥ E, the real number f (x, y) will be denoted by < x, y > or x · ______.