Page 476 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(a)

(b)

Hint Use the Cauchy–Schwarz inequality.

34. (For Readers Who Have Studied Calculus)
Let have the inner product

and let . Show that if , then and are orthogonal with respect to the given inner
product.

35. in an -coordinate system in . Describe the subspace .
(a) Let W be the line

(b) Let W be the y-axis in an -coordinate system in . Describe the subspace .

(c) Let W be the -plane of an -coordinate system in . Describe the subspace .

Let be a homogeneous system of three equations in the unknowns x, y, and z.
36.

(a) If the solution space is a line through the origin in , what kind of geometric object is
the row space of A? Explain your reasoning.

(b) If the column space of A is a line through the origin, what kind of geometric object is the

solution space of the homogeneous system ? Explain your reasoning.

(c) If the homogeneous system has a unique solution, what can you say about the

row space and column space of A? Explain your reasoning.

Indicate whether each statement is always true or sometimes false. Justify your answer by giving
37. a logical argument or a counterexample.