Scalar Product in Euclidean Spaces
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Questions and Answers

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

  • Definite positive and asymmetric
  • Non-symmetric and definite negative
  • Strictly positive coefficients (correct)
  • Rank n and nondegenerate
  • Which of the following properties does a scalar product on E satisfy?

  • Symmetric and bilinear (correct)
  • Nondegenerate and of rank n
  • Positive definite and non-symmetric
  • Definite and asymmetric
  • What is the real number denoted by < x, y > in a scalar product on E?

  • Symmetric and bilinear
  • Definite positive
  • Rank n
  • Value of f (x, y) (correct)
  • What type of form is the canonical scalar product on Rn?

    <p>Symmetric and bilinear</p> Signup and view all the answers

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

    <p>Strictly positive coefficients</p> Signup and view all the answers

    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 = ______.

    <p>0</p> Signup and view all the answers

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

    <p>space</p> Signup and view all the answers

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

    <p>rank</p> Signup and view all the answers

    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

    <p>0</p> Signup and view all the answers

    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 · ______.

    <p>y</p> Signup and view all the answers

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