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(d) part (d )
Prove: If u and v are matrices and A is an matrix, then
27.
Use the Cauchy–Schwarz inequality to prove that for all real values of a, b, and ,
28.
Prove: If , , …, are positive real numbers and if and are any two vectors in
29. , then
Show that equality holds in the Cauchy–Schwarz inequality if and only if u and v are linearly dependent.
30.
Use vector methods to prove that a triangle that is inscribed in a circle so that it has a diameter for a side must be a right
31. triangle.
Figure Ex-31
Hint Express the vectors and in the accompanying figure in terms of u and v.
With respect to the Euclidean inner product, the vectors and have norm 2, and the angle
32. between them is 60°. (see the accompanying figure). Find a weighted Euclidean inner product with respect to which u and
v are orthogonal unit vectors.
Figure Ex-32
33. (For Readers Who Have Studied Calculus)
Let and be continuous functions on [0, 1]. Prove:
