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In Exercises 19 and 20, V is a vector space. Mark each
Chapter 4, Problem 19E(choose chapter or problem)
In Exercises 19 and 20, V is a vector space. Mark each statement True or False. Justify each answer.a. The number of pivot columns of a matrix equals the dimension of its column space.b. A plane in is a two-dimensional subspace of .c. The dimension of the vector space is 4.d. If dim V = n and S is a linearly independent set in V, then S is a basis for V.e. If a set spans a finite-dimensional vector space V and if T is a set of more than p vectors in V, then T is linearly dependent.
Questions & Answers
QUESTION:
In Exercises 19 and 20, V is a vector space. Mark each statement True or False. Justify each answer.a. The number of pivot columns of a matrix equals the dimension of its column space.b. A plane in is a two-dimensional subspace of .c. The dimension of the vector space is 4.d. If dim V = n and S is a linearly independent set in V, then S is a basis for V.e. If a set spans a finite-dimensional vector space V and if T is a set of more than p vectors in V, then T is linearly dependent.
ANSWER:Solution 19EStep 1 of 6The object of the each part is to mark each statement as true or false.(a)Use the result, “The pivot columns of a matrix form a basis for .”So the number of pivot column is same as the number of vectors in basis for Thus, the dimension of it’s the column space is same as the number of pivot columns of .Therefore, the given statement is .