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)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.2: Vector Spaces

Problem 43E: Prove that in a given vector space V, the zero vector is unique.

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Question

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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