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Exercise 1. Answer True or False. i) If u, v are two vectors in the inner product space V such that |u + v|| = ||u|| + ||| then (u, v) < 0. ii) If T : V →V is an operator on the inner product space V such that ||T(u)|| < 2|||| for all u e V, then |A| < 2, for all eigenvalues A of T. iii) Suppose u, v are two non zero vectors in a real inner product space V, if ||u|| = ||||. then u + v is orthogonal to u- v. iv) Consider R2 with its Euclidean inner product. There exists three non-zero vectors in R?, which are mutually orthogonal. v) The function that takes (r1, 12), (Yı, 42) E R? to r12 + I2Yı is an inner product on R?.

Exercise 1. Answer True or False. i) If u, v are two vectors in the inner product space V such that |u + v|| = ||u|| + ||| then (u, v) < 0. ii) If T : V →V is an operator on the inner product space V such that ||T(u)|| < 2|||| for all u e V, then |A| < 2, for all eigenvalues A of T. iii) Suppose u, v are two non zero vectors in a real inner product space V, if ||u|| = ||||. then u + v is orthogonal to u- v. iv) Consider R2 with its Euclidean inner product. There exists three non-zero vectors in R?, which are mutually orthogonal. v) The function that takes (r1, 12), (Yı, 42) E R? to r12 + I2Yı is an inner product on R?.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.1: Vector In R^n
Problem 61E: Illustrate properties 110 of Theorem 4.2 for u=(2,1,3,6), v=(1,4,0,1), w=(3,0,2,0), c=5, and d=2....
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Exercise 1. Answer True or False. i) If u, v are two vectors in the inner product space V such that ||u + v|| = ||| + ||v||, then (u, v) < 0. %3D ii) If T : V → V is an operator on the inner product space V such that ||T(u)|| < 2|||| for all u e V, then |A < 2, for all eigenvalues A of T. iii) Suppose u, v are two non zero vectors in a real inner product space V, if ||u|| = ||||. then u + v is orthogonal to u- v. iv) Consider R? with its Euclidean inner product. There exists three non-zero vectors in R?, which are mutually orthogonal. v) The function that takes (71, x2), (y1, Y2) E R² to r142 + 12Y1 is an inner product on R?.
Transcribed Image Text:Exercise 1. Answer True or False. i) If u, v are two vectors in the inner product space V such that ||u + v|| = ||| + ||v||, then (u, v) < 0. %3D ii) If T : V → V is an operator on the inner product space V such that ||T(u)|| < 2|||| for all u e V, then |A < 2, for all eigenvalues A of T. iii) Suppose u, v are two non zero vectors in a real inner product space V, if ||u|| = ||||. then u + v is orthogonal to u- v. iv) Consider R? with its Euclidean inner product. There exists three non-zero vectors in R?, which are mutually orthogonal. v) The function that takes (71, x2), (y1, Y2) E R² to r142 + 12Y1 is an inner product on R?.
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