True or False? In Exercises 73 and 74, determine whether the each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text.
(a) The length or norm of a vector is
(b) The dot product of two
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Elementary Linear Algebra (MindTap Course List)
- True or false? In Exercises 43and 44, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The orthogonal complement of Rn is empty set. (b) If each vector vRn can be uniquely written as a sum of a vector s1 from S1 and a vector s2 from S2, then Rn is direct sum of S1 and S2.arrow_forwardTrue or False? In Exercises 79 and 80, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows that the statement is not true in all cases or cite an appropriate statement from the text. a If dim(V)=n, then any set of n+1 vectors in V must be linearly dependent. b If dim(V)=n, then any set of n1 vectors in V must be linearly independent.arrow_forwardTrue or False? In Exercises 57and 58, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) To subtract two vectors in Rn, subtract their corresponding components. (b) The zero vector 0 in Rn is the additive inverse of a vector.arrow_forward
- True or False? In Exercises 55 and 56, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a A set S of vectors in an inner product space V is orthonormal when every vector is a unit vector and each pair of vectors is orthogonal. b If a set of nonzero vectors S in an inner product space V is orthogonal, then S is linearly independent.arrow_forwardFinding the Length of a Vector. In Exercises 1-4, find the length of the vector. v=(0,1)arrow_forwardTrue or False? In Exercises 57and 58, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) Two vectors in Rn are equal if and only if their corresponding components are equal. (b) The vector v is the additive identity of v.arrow_forward
- Finding a VectorIn Exercises 13-16, find the vector v with the given length and the same direction as u. v=3,u=(0,2,1,1)arrow_forwardExercises Finding a Unit Vector. In Exercises 912, find a unit vector a in the direction of u and b in the direction opposite that of u. Verify that each vector has length 1. u=(2,2)arrow_forwardTrue or false?In Exercises 85 and 86, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a The dot product is the only inner product that can be defined in Rn. b A nonzero vector in an inner product can have a norm of zero.arrow_forward
- True or False? In Exercises 83-86, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The standard operations in Rn are vector addition and scalar multiplication. (b) The additive inverse of a vector is not unique. (c) A vector space consists of four entities: a set of vectors, a set of scalars, and two operations.arrow_forwardTrue or False? In Exercises 9598, determine whether the statement is true or false. Justify your answer. If u is a unit vector in the direction of v, then v=vu.arrow_forward
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