Problem 1PP: Let u1= [1/52/5] and u2= [2/51/5]. Show that {u1. u2} is an orthonormal basis for Problem 2PP: Let y and L be as in Example 3 and Figure 3. Compute the orthogonal projection of y onto L using u... Problem 3PP: Let U and x be as in Example 6. and let y = [326]. Verify that UxUy = xy EXAMPLE 6 Let U =... Problem 4PP: Let U be an n n matrix with orthonormal columns. Show that det U = 1. Problem 1E: In Exercises 16, determine which sets of vectors are orthogonal. 1. [143], [521], [347] Problem 2E: In Exercises 16, determine which sets of vectors are orthogonal. 2. [121], [012], [521] Problem 3E: In Exercises 16, determine which sets of vectors are orthogonal. 3. [271], [639], [311] Problem 4E Problem 5E: In Exercises 16, determine which sets of vectors are orthogonal. 5. [3213], [1334], [3870] Problem 6E: In Exercises 16, determine which sets of vectors are orthogonal. 6. [5403], [4138], [3351] Problem 7E: In Exercises 710, show that {u1, u2} or {u1, u2, u3} is an orthogonal basis for 2 or 3, respectively... Problem 8E Problem 9E: In Exercises 710, show that {u1, u2} or {u1, u2, u3} is an orthogonal basis for 2 or 3, respectively... Problem 10E: In Exercises 710, show that {u1, u2} or {u1, u2, u3} is an orthogonal basis for 2 or 3, respectively... Problem 11E: Compute the orthogonal projection of [17] onto the line through [42] and the origin. Problem 12E Problem 13E: Let y = [23] and u = [47] Write y as the sum of two orthogonal vectors, one in Span {u} and one... Problem 14E: Let y=26 and u=61. Write y as the sum of a vector in Span u and a vector orthogonal to u. Problem 15E: Let y = [31] and u = [86] Compute the distance from y to the line through u and the origin. Problem 16E: Let y = [39] and u = [12] Compute the distance from y to the line through u and the origin. Problem 17E: In Exercises 1722, determine which sets of vectors are orthonormal. If a set is only orthogonal,... Problem 18E: In Exercises 1722, determine which sets of vectors are orthonormal. If a set is only orthogonal,... Problem 19E: In Exercises 1722, determine which sets of vectors are orthonormal. If a set is only orthogonal,... Problem 20E: In Exercises 1722, determine which sets of vectors are orthonormal. If a set is only orthogonal,... Problem 21E: In Exercises 1722, determine which sets of vectors are orthonormal. If a set is only orthogonal,... Problem 22E: In Exercises 1722, determine which sets of vectors are orthonormal. If a set is only orthogonal,... Problem 23E: In Exercises 23—32, all vectors are in Rn. Mark each statement True or False (T/F). Justify each... Problem 24E Problem 25E Problem 26E Problem 27E Problem 28E Problem 29E Problem 30E Problem 31E Problem 32E Problem 33E: Prove Theorem 7. [Hint: For (a), compute |Ux||2, or prove (b) first] Problem 34E: Suppose W is a sub space of n spanned by n nonzero orthogonal vectors. Explain why W = n. Problem 35E: Let U be a square matrix with orthonormal columns. Explain why U is invertible. (Mention the... Problem 36E: Let U be an n n orthogonal matrix. Show that the rows of U form an orthonormal basis of n. Problem 37E: Let U and V be n n orthogonal matrices. Explain why UV is an orthogonal matrix [That is, explain... Problem 38E: Let U be an orthogonal matrix, and construct V by inter-changing some of the columns of U. Explain... Problem 39E: Show that the orthogonal projection of a vector y onto a line I. through the origin in 2 does not... Problem 40E: Let {v1, v2} be an orthogonal set of nonzero vectors, and let c1, c2 be any nonzero scalars. Show... Problem 41E Problem 42E: Given u 0 in n, let L = Span{u}. For y in n, the reflection of y in L is the point reflL y defined... Problem 43E Problem 44E: In parts (a)—(d), let U be the matrix formed by normalizing each column of the matrix A in... format_list_bulleted